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BURR,  51  J;OHW  ST, 


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LIBRARY 

OF   THK 

UNIVERSITY  OF  CALIFORNIA 

OR 


Received 
Accession  No  .  (o  7  ^  Q../....     •    Class  No. 


PRACTICAL 

-7;* 


ARITHMETIC: 


EMBRACING 


THE  SCIENCE  AND  APPLICATIONS  OF  NUMBERS. 


BY 

CHARLES  DA  VIES,  LL.  D., 

P^X)FESSOR    OF    HIGHER    MATHEMATICS    IN    COLUMBIA    COLLEGE  J     AND 
AUTHOR   OF   DIFFERENTIAL   CALCULUS,    ANALYTICAL   GEOMETRY, 
DESCRIPTIVE    GEOMETRY,    ELEMENTS    OF    SURVEYING, 
AND    ARITHMETICS. 


NEW  YORK: 
PUBLISHED   BY   BARNES   &   BURR, 

51   &  53   JOHN   STREET. 
1863. 


ADVEKTISEMENT. 


THE  attention  of  Teachers  is  respectfully  invited  to  the  REVIRJ 
EDITIONS  of 

ies'  Arithmetical  Juries 

FOR  SCHOOLS  AND  ACADEMIES. 


1.  DAVIES'  PRIMARY  ARITHMETIC. 

2.  DAVIES'  INTELLECTUAL  ARITHMETIC. 

3.  DAVIES'  PRACTICAL  ARITHMETIC. 

4.  DAVIES'  UNIVERSITY  ARITHMETIC. 

5.  DAVIES'  PRACTICAL  MATHEMATICS. 


The  above  Works,  by  CHARLES  DAVIES,  LL.  D.,  Author  of  a  Com- 
plete Course  of  Mathematics,  are  designed  as  a  full  Course  of  Arith- 
metical Instruction  necessary  for  the  practical  duties  of  business 
life  ;  and  also  to  prepare  the  Student  for  the  more  advanced  Series 
of  Mathematics  by  the  same  Author. 

The  following  New  Editions  of  Algebra,  by  Professor  DAVIES,  are 
commended  to  the  attention  of  Teachers : 

1.  DAVIES'  NEW  ELEMENTARY  ALGEBRA  AND  KEY. 

2.  DAVIES'  UNIVERSITY  ALGEBRA  AND  KEY. 

3.  DAVIES'  BOURDON'S  ALGEBRA  AND  KEY. 


Entered  according  to  Act  of  Congress,  in  the  year  one  thousand  eight  hundred 
and  sixty-two, 

BY    CHARLES    DAVIES, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern 
District  of  New  York. 


KWJ*E] 


PREFACE. 


ARITHMETIC  embraces  the  science  of  numbers,  together  with  aft 
the  rules  which  are  employed  in  applying  the  principles  of  the 
science  to  practical  purposes.  It  is  the  foundation  of  the  exact 
and  mixed  sciences,  and  the  first  subject,  in  a  well-arranged  course 
of  instruction,  to  which  the  reasoning  powers  of  the  mind  are 
directed.  Because  of  its  great  uses  and  applications,  it  has  be- 
come the  guide  and  daily  companion  of  the  mechanic  and  man 
of  business.  In  the  present  work,  a  few  general  principles  are 
laid  down,  to  which  all  the  operations  in  numbers  may  be  re- 
ferred : 

1st.  The  unit  1  is  regarded  as  the  base  of  every  number,  and 
the  consideration  of  it  is  the  first  step  in  the  analysis  of  every 
question  relating  to  numbers. 

2d.  Every  number  is  treated  as  a  collection  of  units,  or  as 
made  up  of  sets  of  such  collections ;  each  collection  having  its 
own  base,  which  is  either  1,  or  some  number  derived  from  1. 

3d.  The  number  expressing  the  relation  between  two  different 
units  of  a  number,  is  called  the  SCALE  ;  and  the  employment  of 
this  term  enables  us  to  generalize  the  laws  which  regulate  the 
formation  of  numbers. 

4th.  By  employing  the  term  "fractional  unit"  the  same  prin- 
ciples are  made  applicable  to  fractional  numbers;  for  all  frac- 
tions are  but  collections  of  fractional  units,  these  units  having 
a  known  relation  to  1. 

5th.  The  presentation  of  the  fractional  units  to  the  minds  of 
young  pupils,  by  means  of  a  diagram,  as  exhibited  in  the  Primary 
and  Intellectual  Arithmetics,  has  greatly  simplified  the  operations 
in  fractions;  and  they  may  now  be  placed  before  Denominate 
Numbers,  where,  in  a  purely  scientific  arrangement,  they  properly 
belong. 


4  PREFACE. 

In  the  preparation  of  the  work,  two  objects  have  been  kept 
'  constantly  in  view : 

1st.  To  make  it  Educational ;    and, 
2d.    To  make  it  Practical. 

To  attain  these  ends,  the  following  plan  has  been  adopted : 

1.  To  introduce  every  new  idea  to  the  mind  of  the  pupil  by 
a  simple  question,  and  then  to  express  that  idea  in  general  terms 
under  the  form  of  a  definition. 

2.  When  a  sufficient  number  of  ideas  are  thus  fixed  in  the 
mind,  they  are  combined  to  form  the  basis  of  an  analysis;  so 
that  all  the  principles  are  developed  by  analysis  in  their  proper 
order. 

3.  The  work  has  been  divided  into  sections,  each  containing  a 
number  of  connected  principles;  and  these  sections  constitute  a 
series  of  dependent  propositions  that  make  up  the  entire  system 
of  principles  and  rules  which  the  work  develops. 

Great  pains  have  been  taken  to  make  the  work  PRACTICAL  in 
its  general  character,  by  explaining  and  illustrating  the  various 
applications  of  Arithmetic  in  the  transactions  of  business,  and 
by  connecting,  as  closely  as  possible,  every  principle  or  rule,, 
with  all  the  applications  which  belong  to  it. 

I  have  great  pleasure  in  acknowledging  my  obligations  to 
many  teachers  who  have  favored  me  with  valuable  suggestions  in 
regard  to  definitions,  rules,  and  methods  of  illustration.  Their 
generous  appreciation  of  my  labors  has  been  both  an  encourage- 
ment and  a  reward. 


COLUMBIA  COLLEGE,  NEW  YOKK, 
January,  1863. 


CONTENTS. 


FIRST    FIVE    RULES. 

Definitions,                            ...  9 

Notation  and  Numeration,                  .                                          .  10 

Addition  of  Simple  Numbers,                                              .        .  22 

Applications  in  Addition,            30 

Subtraction  of  Simple  Numbers, 33 

Applications  in  Subtraction, 37 

Multiplication  of  Simple  Numbers, 41 

Composite  Numbers — Factors,            47 

Contractions  in  Multiplication, 48 

Applications, 51 

Division  of  Simple  Numbers,             . 54 

Proof  of  Multiplication, 62 

Contractions  in  Division,             63 

Applications  in  the  preceding  Rules, 67 

PROPERTIES    OF    NUMBERS. 

Exact  Divisors,             72 

Composite  and  Prime  Numbers, 72 

Cancellation, 74 

Least  Common  Multiple,             77 

Greatest  Common  Divisor, 79 

COMMON    FRACTIONS. 

Definitions, 82 

Properties  of  Fractions,       ...  '  .  .85 

Different  kinds  of  Fractions, .86 

Six  Fundamental  Propositions, 87 

Reduction  of  Fractions, .'90 

Addition  of  Fractions, 98 

Subtraction  of  Fractions,            ....*.                .  100 

Multiplication  of  Fractions,        .                 102 

Division  of  Fractions,          .                         107 

Complex  Fractions,              .         .                 .        .  Ill 

Miscellaneous  Examples,             .                                                    .  112 


6  CONTENTS. 

DECIMAL   FRACTIONS.  PAQ« 

Definition  of  Decimal  Fractions,      .                         ...  114 

Decimal  Numeration, 114 

Annexing  and  Prefixing  Ciphers,    .        .        .        .        .        .  118 

Addition  of  Decimals,        .        .                119 

Subtraction  of  Decimals, 122 

Multiplication  of  Decimals, 123 

Division  of  Decimals, 125 

Applications  in  the  Four  Rules, 128 

UNITED    STATES    CURRENCY. 

United  States  Currency, 130 

Reduction  of  Currency, 131 

Addition  of  United  States  Currency,       ...                .  132 

Subtraction  of  United  States  Currency,           ....  135 

Multiplication  of  United  States  Currency,      .        .                .  136 

Division  of  United  States  Currency, 138 

Applications  in  the  Four  Rules,      .        .        .        .        .        .  139 

DENOMINATE    NUMBERS. 

Definitions, 147 

Different  kinds  of  Units, 147 

Abstract  Numbers,             147 

United  States  Currency,            148 

English  Currency, 149 

Reduction  Descending, 150 

Reduction  Ascending, 150 

Long  Measure, 151 

Surveyors'  Measure, 153 

Cloth  Measure, 154 

Square  Measure, 155 

Cubic  Measure,  or  Measure  of  Volume,          ....  157 

Liquid  Measure, 159 

Dry  Measure,                      160 

Avoirdupois  Weight, 161 

Troy  Weight,             163 

Apothecaries'  Weight, 164 

Measure  of  Time, 165 

Circular  Measure,  or  Motion, 168 

Miscellaneous  Tables, 169 

Miscellaneous  Examples, 170 


CONTENTS.  7 

PACK 

DENOMINATE  FRACTIONS,         .  .172 

Addition  of  Compound  Numbers,             .        .                         .  178 

Addition  of  Fractions,       ...                ....  182 

Subtraction  of  Compound  Numbers, 183 

Time  between  Dates, 185 

Subtraction  of  Fractions, 187 

Multiplication  of  Compound  Numbers,            .        .        .        .  188 

Division  of  Compound  Numbers, 190 

Applications, 193 

Longitude  and  Time, 195 

DUODECIMALS,           198 

Addition  and  Subtraction, 199 

RATIO    AND    PROPORTION. 

Ratio  denned — Simple  and  Compound,            .        .        .  202 

Simple  Proportion,             205 

Compound  Proportion, 207 

Rule  of  Three, 208 

Double  Rule  of  Three, 213 

APPLICATIONS.  •> 

Partnership, 216 

Percentage, 219 

Commission, 223 

Profit  and  Loss, '227 

Insurance, 231 

Stocks  and  Brokerage, 232 

Interest,              235 

Partial  Payments, 241 

Problems  in  Interest, .  244 

Compound  Interest, 245 

Discount,            248 

Banking,             250 

Exchange, 253 

Equation  of  Payments, 257 

Assessing  Taxes,        ......                         .  262 

Custom-house  Business,            . 264 

Currency,            .        .                266 

Analysis,             .        .                                                        .        .  267 


8  CONTENTS. 

PAGE 

ALLIGATION  MEDIAL,        .               .       .  275 

ALLIGATION  ALTERNATE,         ...                      .  276 

INVOLUTION. 

Definition  of,  &c.,       ....                .  280 

EVOLUTION. 

Definition  of,  &c., .281 

Extraction  of  the  Square  Boot,                .        .        .        .        .  281 

Applications  in  Square  Root, 286 

Extraction  of  the  Cube  Root, 289 

Applications  in  Cube  Root,      .        .        .        .^              .  293 

ARITHMETICAL   PROGRESSION. 

Definition  of,  &c., .294 

Different  Cases,          . •  295 

GEOMETRICAL   PROGRESSION. 

Definition  of,  &c.,  298 

Cases,  299 

MENSURATION. 

To* find  the  area  of  a  Triangle,        .                         ...  302 

To  find  the  area  of  a  Square,  Rectangle,  &c.,        .        .        .  303 

To  find  the  area  of  a  Trapezoid, 303 

To  find  the  circumference  and  diameter  of  a  Circle,     .        .  304 

To  find  the  area  of  a  Circle, 304 

To  find  the  surface  of  a  Sphere,     ...                .        .  305 

To  find  the  contents  of  a  Sphere, 305 

To  find  the  convex  surface  of  a  Prism,          .        .                .  306 

To  find  the  contents  of  a  Prism,     .                         .        .  306 

To  find  the  convex  surface  of  a  Cylinder,      ....  307 

To  find  the  contents  of  a  Cylinder,, 307 

To  find  the  contents  of  a  Pyramid,        ....  308 
To  find  the  contents  of  a  Cone,       ....                .309 

GAUGING. 

Rules  for  Gauging,            .                                .  310 

PROMISCUOUS  EXAMPLES,                              ....  311 


ARITHMETIC. 


Definitions, 

1.  A  UNIT   is  a  single  thing,  or  one. 

2.  QUANTITY  is  any  thing  which  can  be  measured  by  a  unit. 

3.  A  NUMBER   is  a  unit,  or  a  collection  of  units. 

4.  An  ABSTRACT  NUMBER  is  one  whose  unit  is  not  named ; 
as,  one,  two,  three,  &c. 

5.  A  DENOMINATE  NUMBER    is  one  whose  unit  is  named  ; 
as,   one  foot,   two  yards,   three  pounds,   &c.     Such  numbers 
are  also  called,  Concrete  numbers. 

6.  A  SIMPLE  NUMBER    is  a  single  collection  of  like  units, 
whether  abstract  or  denominate. 

7.  ARITHMETIC    is  the   Science  of  numbers,  and  also,  the 
Art  of  applying  numbers  to  practical  purposes. 

8.  A  PROPOSITION    is   something   to   be   done,  or  demon- 
strated. 

9.  An  ANALYSIS    is  an  examination  of  the  separate  parts 
of  a  proposition. 

10.  An  OPERATION    is   the   act   of  doing   something  with 
numbers. 

11.  A  RULE  is  the  direction  for  performing  an  operation. 

12.  An  ANSWER   is  the  result  of  a  correct  operation. 


1.  What  is  a  unit?— 2.  What  is  quantity?— 3.  What  is  a  num- 
ber ? — 4.  What  is  an  abstract  number  ? — 5.  What  is  a  denominate 
number  ?  What  other  name  has  it  ? — 6.  What  is  a  simple  num- 
ber?—7.  What  is  Arithmetic?— 8.  What  is  a  proposition ?— 9  What 
is  an  analysis? — 10.  What  is  an  operation? — 11.  What  is  a  rule? 
— 12.  What  is  an  answer? 

1* 


10  NOTATION  AND   NUMERATION. 

Operations  of  Arithmetic. 

13.  There  are,  in  -Arithmetic,  five  fundamental  operations: 
Notation  and  Numeration,  Addition,  Subtraction,  Multiplica- 
tion, and  Division. 

Expressing  Numbers. 

14.  There  are  three  methods  of  expressing  numbers : 

1.  By  words,  or  common  language,  spoken  or  written. 

2.  By  capital  letters;  called,  the  Roman  method. 

3.  By  figures  ;  called,  the  Arabic  method. 

Expressing  Numbers  by  Words. 

15.  A  single  thing  is  called One. 

One  and  one  more, Two. 

Two  and  one  more, Three. 

Three  and  one  more, Four. 

Four  and  one  more, Five. 

Five  and  one  more,  ...  .  Six. 

Six  and  one  more, Seven. 

Seven  and  one  more, Eight. 

Eight  and  one  more, Nine. 

Nine  and  one  more, Ten. 

&c.,  &c. 

Each  of  the  words,  one,  two,  three,  four,  five,  six,  &c., 
expresses  a  number,  and  denotes  how  many  units  are  taken. 


NOTATION  AND   NUMERATION. 

16.  NOTATION  is  the  method  of  expressing  numbers,  either 
by  letters  or  figures. 

NUMERATION  is  the  art  of  reading,  correctly,  any  number 
expressed  by  letters  or  figures. 

There  are  two  methods  of  Notation :  the  one  by  letters,  the 
other  by  figures.  The  method  by  letters  is  called,  the  Roman 
Notation  ;  the  method  by  figures  is  called,  the  Arabic  Notation, 


NOTATION   AND   NUMERATION. 


11 


Roman  Notation. 

17.    The   Roman   notation    employs    seven    capital  letters. 
They  express  the  following  values : 

I         Y         X          L          C          D          M 

One,          five,  ten,          fifty,       ^d^  hundred,  ^0°"^. 

All  other  numbers  are  expressed  by  combining  these  let- 
ters, according  to  the  following  principles : 

1.  Every  time  a  letter  is  repeated,  the  number  which  it 
denotes  is  repeated. 

2.  If  a  letter  denoting  a  less  number  be  written  on  the 
right  of  one  denoting  a  greater,  the  number  expressed  will  be 
denoted  by  the  sum  of  the  numbers. 

3.  If  a  letter  denoting  a  less  number  be  written  on  the 
left  of  one  denoting  a  greater,  the  number  expressed  will  be 
the  difference  of  the  numbers. 

4.  A  dash  ( — ),  placed  over  a  letter,  increases  the  number 
for  which  it  stands,  a  thousand  times. 


Roman  Table. 


I 

II 

III 

IY 

Y 

YI 

YII 

VIII 

IX 

X 

XX 

XXX 

XL 

L 

LX 

LXX 


One. 

LXXX 

Two. 

XO     . 

Three. 

0 

Four. 

CO      . 

Five. 

COO  . 

Six. 

ccoo 

Seven. 

D 

Eight. 

DC     . 

Nine. 

DOC  . 

Ten. 

DCCO 

Twenty. 

DCCCC 

Thirty. 

M-      . 

Forty. 

MD     . 

Fifty: 

MM    . 

Sixty. 

Y 

Seventy. 

X 

Eighty. 
Ninety. 
One  hundred. 
Two  hundred. 
Three  hundred. 
Four  hundred. 
Five  hundred. 
Six  hundred. 
Seven  hundred. 
Eight  hundred. 
Nine  hundred. 
One  thousand. 
Fifteen  hundred. 
Two  thousand. 
Five  thousand. 
Ten  thousand. 


12  NOTATION   AND   NUMERATION. 

Examples  in  Roman  Notation. 

Express  the  following  numbers  by  letters  : 

1.  Fifteen. 

2.  Nineteen. 

3.  Twenty-nine. 

4.  Thirty-five. 

5.  Forty-seven. 

6.  Ninety-nine. 

7.  One  hundred  and  sixty. 

8.  Four  hundred  and  forty-one. 

9.  Five  hundred  and  sixty-nine. 

10.  One  thousand  one  hundred  and  six. 

11.  Two  thousand  and  twenty-five. 

12.  Six  hundred  and  ninety-nine. 

13.  One  thousand  nine  hundred  aud  twenty-five. 

14.  Two  thousand  six  hundred  and  eighty. 

15.  Four  thousand  nine  hundred  and  sixty-five. 

16.  Two  thousand  seven  hundred  and  ninety-one. 

17.  One  thousand  nine  hundred  and  sixteen. 

18.  Two  thousand  six  hundred  and  forty-one. 

19.  One  thousand  eight  hundred  and  sixty-two. 

20.  Twenty  thousand  five  hundred  and  twelve. 


13.  How  many  fundamental  operations  are  there  in  Arithmetic? 
Name  them. 

14.  How  many  methods  are  there  of  expressing  numbers  ?    WLat 
are  they? 

15.  What  is  a  single  thing  called  ?    One  and  one  more  ?     Six  and 
one  more?    Eight  and  one  more? 

16.  What    is    Notation  ?     What    is    Numeration  ?     How   many 
methods  of  notation   are  there  ?    What   is   the   Roman  method  ? 
What,  the  Arabic? 

17.  How  many  letters  does  the  Roman  notation  employ  ?    Which 
are  they  ?    What  value  does  each  represent  ?    What  is  the  effect 
of  repeating  a  letter  ?    What  is  the  number,  when  a  letter  denoting 
a  less  number  is  placed  on  the  right  of  one  denoting  a  greater  ? 
What  is  the  number,  when   a  letter  denoting  a  less   number  is 
placed  on  the  left  of  one  denoting  a  greater?    What  is  the  effect 
of  placing  a  dash  over  a  letter  ? 


NOTATION   AND    NUMERATION.  13 

Arabic  Notation. 

18.  ARABIC  NOTATION  is  the  method  of  expressing  num- 
bers by  figures.  Ten  figures  are  used.  They  are, 

0123456789 

Naught,    one,      two,    three,    four,     five,      six,    seven,  eight,    nine. 

These  figures  are  the  Alphabet  of  the  Arabic  Notation. 

The  0  is  called,  naught,  cipher,  or  zero.  It  denotes  no 
number.  Thus,  if  there  are  no  apples  in  a  basket,  we  write, 
the  number  of  apples  in  the  basket  is  0.  The  other  nine 
figures  are  called,  Significant  Figures,  or  Digits. 

Orders  of  Units. 

1,9.  Nine  is  the  highest  number  which  can  be  expressed  by 
a  single  figure.  To  express  ten,  we  write  0  on  the  right  of  1 ; 

Thus,     .       .- 10; 

which  is  read,  ten. 

This  10  is  equal  to  ten  of  the  units  expressed  by  1.  It 
is  but  a  single  ten,  and'  is  a  unit,  the  value  of  which  is  ten 
times  as  great  as  the  unit  one.  It  is  called,  a  unit  of  the 
second  order. 

20.  When  two  figures  are  written  by  the  side  of  each  other, 
the  one  on  the  right  is  in  the  place  of  units,  and  the  other 
in  the  place  of  tens,  or  of  units  of  the  second  order.  Each 
unit  of  the  second  order  is  equal  to  ten  units  of  the  first  order. 
When  units  simply  are  named,  units  of  the  first  order  are 
always  meant. 

18.  What  is   Arabic  Notation  ?     How  many  figures  are  used  ? 
Name  the  figures.     What  do  they  form  ?     How  many  things  does 
1  express  ?    How  many  things  does  5  express  ?    How  many  units 
in  3?    In  7?    In  9?    In  8?    In  0?    What  are  the  figures,  with 
one  exception,  called  ?    Which  are  the  significant  figures  ? 

19.  What  is  the  highest  number  that   can  be   expressed  by  a 
single  figure  ?    How  do  we  express  ten  ?    To  how  many  units  1  is 
ten  equal  ?     May  we  consider  it  a  single  unit  ?    Of  what  order  ? 


4  NOTATION   AND   NUMERATION.. 

Units  of  the  second  order  are  written  thus : 

One  ten,  or         ...              .       .   *   .       .  10. 

Two  tens,  or  twenty, 20. 

Three  tens,  or  thirty,      ......  30. 

Four  tens,  or  forty,         .       ,       .       .       .       .40. 

Five  tens,  or  fifty,    .......  50. 

Six  tens,  or  sixty, 60. 

Seven  tens,  or  seventy, 70. 

Eight  tens,  or  eighty,                   .  80. 

Nine  tens,  or  ninety, 90. 

The  intermediate  numbers  between  10  and  20,  between  20 
and  30,  &c.,  may  be  expressed  by  considering  their  tens  and 
units.  For  example,  the  number  twelve  is  made  up  of  one 
ten  and  two  units.  It  is  written  by  setting  1  in  the  place 
of  tens,  and  2  in  the  place  of  units ; 

Thus, 12. 

Eighteen,  has  1  ten  and  8  units,       .  .  .18. 

Twenty-five,  has  2  tens  and  5  units, .  .  .25. 

Thirty-seven,  has  3  tens  and  7  units,  .  .     37. 

Fifty-four,  has  5  tens  and  4  units,     .  .  .54. 

Ninety-nine,  has  9  tens  and  9  units,  .  .  .99. 

Hence,  any  number  greater  than  nine,  and  less  than  one 
hundred,  may  be  expressed  by  two  figures. 

21.  la  order  to  express  ten  units  of  the  second  order,  or 
one  hundred,  we  form  a  new  combination, 

thus, 100, 

by  writing  two  ciphers  on  the  right  of  1.     This  number  is 
ead,  one  hundred. 

20.  When  two  figures  are  written  by  the  side  of  each  other,  what 
is  the  place  on  the  right  called?  The  place  on  the  left?  When 
units  simply  are  named,  what  units  are  meant  ?  How  many  units 
of  the  second  order  in  20?  In  30?  In  40?  In  50?  In  60?  In 
70  ?  In  80  ?  In  90  ?  Of  what  is  the  number  12  made  up  ?  Also, 
18,  25,  37,  54,  99  ?  What  numbers  may  be  expressed  by  two  figures  1 


•  NOTATION  AND  NUMERATION.  1 

This  one  hundred  expresses  10  units  of  the  second  order, 
or  100  units  of  the  first  order.  The  one  hundred  is  but  a 
single  hundred,  and  is  a  unit  of  the  third  order. 

We  can  now  express  any  number  less  than  one  thousand. 

For  example,  in  the  number  three  hundred  and 
seventy-five,  there  are  5  units,  7  tens,  and  3  hun-  ?§   «  £ 

dreds.     Write,  therefore,  5  units  of  the  first  order,  7          J  5   § 
of  the  second  order,  and  3  of  the  third;  and  read  375 

from  the  left,  three  hundred  and  seventy-five. 

In  the    number   eight   hundred    and    ninety-nine,  g   ^  -2 

there  are  9  units  of  the  first  order,  9  of  the  second,  jg  %   § 

and  8  of  the  third;   it  is  read,  eight  hundred  and  899 
ninety -nine. 

In  the  number  four  hundred  and  six,  there  are  g   »  5 

6  units  of  the  first  order,  0  of  the  second,  and  4  of          js  5   § 
the  third.  406 

Hence,  we  have  the  following  law  of  the  units : 

The  right-hand  figure  always  expresses  units  of  the  first 
order ;  the  second,  units  of  the  second  order  ;  and  the  third, 
units  of  the  third  order. 

22.  To  express  ten  units  of  the  third  order,  or  one  thousand, 
we  form  a  new  combination  by  writing  three  ciphers  on  the 
right  of  1 ; 

Thus,         .........     1000. 

This  is  but  one  single  thousand,  and  is  a  unit  of  the  fourth 
order. 


21.  How  do  you  write  one  hundred  ?  To  how  many  units  of  the 
second  order  is  it  equal  ?  To  how  many  of  the  first  order  ?  May 
it  be  considered  a  single  unit  ?  Of  what  order  is  it  ?  How  many 
units  of  the  third  order  in  200?  In  300?  In  400?  In  500?  In 
600?  Of  what  is  the  number  375  composed?  The  number  899? 
The  number  406  ?  What  numbers  may  be  expressed  by  three 
figures  ?  What  order  of  units  will  each  figure  express  ? 


16  NOTATION   AND   NUMERATION. 

Thus,  we  may  form  as  many  orders  of  units  as  we  please : 

A  unit  of  the  first  order  is  expressed  by     .  •     .  .          1. 

A  unit  of  the  second  order,  by  1  and  0;  thus,  .         10. 

A  unit  of  the  third  order,  by  1  and  two  O's,     .  .       100. 

A  unit  of  the  fourth  order,  by  1  and  three  O's,  .     1000. 

A  unit  of  the  fifth  order,  by  1  and  four  O's,     .  .  10000. 

And  so  on,  for  units  of  higher  orders. 

23.   Therefore, 

1st.  The  same  figure  expresses  different  units,  according 
to  the  place  it  occupies. 

2d.  Units  of  the  first  order  occupy  the  place  on  the  right ; 
units  of  the  second  order,  the  second  place  ;  units  of  the  third 
order,  the  third  place ;  and  the  unit  of  every  figure  is  deter- 
mined by  the  number  of  its  place. 

3d.  Ten  units  of  the  first  order  make  one  of  the  second ; 
ten  of  the  second,  one  of  the  third ;  ten  of  the  third,  one  of 
the  fourth ;  and  so  on,  for  the  higher  orders. 

4th.  When  figures  are  written  by  the  side  of  each  other, 
ten  units  in  any  place  make  one  unit  of  the  place  next  to 
the  left. 

22.  To  what  are  ten  units  of  the  third  order  equal  ?    How  do  you 
write  it  ?    How  is  a  unit  of  the  first  order  written  ?    How  do  you 
write  a  unit  of  the  second  order  ?    One  of  the  third  ?    One  of  the 
fourth  ?    One  of  the  fifth  ? 

23.  On  what  does  the  unit  of  a  figure  depend  ?    What  is  the 
unit  of  the  first   place  on   the   right  ?    What  is  the  unit   of  the 
second  ^>lace?    What    is   the    unit   of   the   third   place?     Of   the 
fourth  ?    Of  the  fifth  ?     Sixth  ?    How  many  units  of  the  first  order 
make   one  of  the  second  ?     How  many  of  the  second  one  of  the 
third?     How  many  of  the  third  one  of  the  fourth?   &c.     When 
figures  are  written  by  the  side  of  each  other,  how  many  units  of 
any  place  make  one  unit  of  the  place  next  to  the  left  ?     To  how 
many  units  is  1  hundred  equal  ?     To  how  many  tens  ?     To  how 
many  tens  is  1  thousand  equal  ?     To  how  many  hundreds  ?     To 
how  many  units  of  the  first  order  is  one  unit  of  the  third  order 
equal  ?     To  how  many  of  second  order  ? 


NOTATION   AND   NUMERATION.  17 

Examples  in  writing  the  Orders  of  Units. 

1.  Write  3  tens. 

2.  Write  8  units  of  the  second  order. 

3.  Write  9  units  of  the  first  order. 

4.  Write  4  units  of  the  first  order,  5  of  the  second,  6  of 
the  third,  and  8  of  the  fourth. 

5.  Write  9  units  of  the  fifth  order,  none  of  the  fourth,  8 
of  the  third,  7  of  the  second,  and  6  of  the  first. 

6.  Write  one  unit  of  the  sixth  order,  5  of  the  fifth,  4  of 
the  fourth,  9  of  the  third,  7  of  the  second,  and  0  of  the 
first. 

7.  Write  9  units  of  the  5th  order,  0  of  the  4th,  8  of  the 
3d,  1  of  the  2d,  and  3  of  the  1st. 

8.  Write  7  units  of  the  6th  order,  8  of  the  5th,  0  of  the 
4th,  5  of  the  3d,  7  of  the  2d,  and  1  of  the  1st. 

9.  Write  9  units  of  the  7th  order,  0  of  the  6th,  2  of  the 
5th,  3  of  the  4th,  9  of  the  3d,  2  of  the  2d,  and  9  of  the  1st. 

10.  Write  8  units  of  the  8th  order,  6  of  the  7th,  9  of  the 
6th,  8  of  the  5th,  1  of  the  4th,  0  of  the  3d,  2  of  the  2d, 
and  8  of  the  1st. 

11.  Write  14  units  of  the  12th  order,  with  9  of  the  10th, 
6  of  the  8th,  7  of  the  6th,  6  of  the  5th,  5  of  the  3d,  and  3 
of  the  first. 

12.  Write  13  units  of  the  13th  order,  8  of  the  12th,  7  of 
the  9th,  6  of  the  8th,  9  of  the  7th,  7  of  the  6th,  3  of  the 
4th,  and  9  of  the  first. 

13.  Write  9  units  of  the  18th  order,  7  of  the  16th,  4  of 
the  15th,  8  of  the  12th,   3  of  the  llth,   2  of  the  10th,  1 
of  the  9th,  0  of  the  8th,  6  of  the  7th,  2  of  the  third,  and 
1  of  the  1st. 

14.  Write  8  units  of  the  8th  order,  6  of  the  7th,  9  of 
the  6th,  8  of  the  5th,  1  of  the  4th,  0  of  the  3d,  2  of  the 
2d,  and  8  of  the  1st. 

15.  Write  1  unit  of  the   9th  order,   6  of  the   8th,   9  of 
the  7th,  7  of  the  6th,  6  of  the  5th,  5  of  the  4th,  4  of  the 
3d,  3  of  the  2d,  and  2  of  the  1st. 


18  NOTATION   AND   NUMERATION. 

Numeration  Table. 

Cth  Period.    5th  Period.    4th  Period.     3d  Period.     2d  Period.     1st  Period. 
Quadrillions.    Trillions.        Billions.        Millions.      Thousands.        Units. 


s  of  Quadrillions. 
Quadrillions 
ions  . 

[s  of  Trillions  . 
Trillions  .  . 

• 

[s  of  Billions 

i 

* 

Ls  of  Millions 

Millions  .  .  . 

Is  of  Thousands  . 
Thousands  . 

•3 

• 

• 

1*1 

»  <£ 

02 

g 

1 

<g 

d 

1 

C+H      02 

o  d 

go 

1 

'o 

la| 

«£<§ 

ll 

1 

TJ 

d 

CO 

rt 

o 

w 

o 

d    r^ 

H  § 

ll 

o 

1 

W 

II 

t 

0 

( 

4 

§ 

6. 

g 

( 

7  5. 

. 

t 

8 

7  9. 

f 

, 

, 

. 

6, 

0 

2   3. 

. 

. 

§ 

. 

8 

2, 

3 

0    1. 

t 

. 

, 

, 

1    2 

3, 

0 

8   7. 

t 

t 

g 

7, 

0   0 

o, 

7 

3   5. 

g 

( 

, 

4   3, 

2    1 

0, 

4 

6    0. 

t 

, 

, 

5 

48, 

0   0 

o, 

0 

8   7. 

t 

, 

6, 

2 

4   5, 

2   8 

9, 

4 

2   1. 

f 

, 

7 

2, 

5 

4   9, 

1    3 

6, 

8 

2   2. 

t 

8 

9 

4, 

6 

0   2, 

0   4 

3, 

2 

8   8. 

. 

7, 

6 

4 

1, 

0 

0   0, 

9    0 

7, 

4 

5    6. 

t 

8 

4, 

9 

1 

2, 

8 

7   6, 

4   1 

9, 

2 

8   5. 

t 

9   1 

2, 

7 

6 

1, 

2 

5   7, 

3   2 

7, 

8 

2    6. 

6, 

4  0 

7, 

2 

1 

2, 

9 

3    6, 

8   7 

6, 

5 

4   1. 

5  7, 

2   8 

9, 

6 

7 

8, 

5 

4   1, 

2   9 

7, 

3 

1    3. 

920, 

3   2 

3, 

8 

4 

2, 

7 

6   8, 

3    1 

9, 

6 

7   5. 

1.  Numbers  expressed  by  more  than  three  figures,  are  separated 
into  periods  of  three  figures  each,  beginning  at  the  right,  and  are 
written  and  read  by  periods,  as  shown  in  the  above  table. 

2.  Each  period  contains  three  figures,  except  the  last,  which  may 
contain  one,  two,  or  three  figures. 

3.  The  unit  of  the  first  period  is  1 ;  the  unit  of  the  second  period, 
1  thousand ;  of  the  third,  1  million ;  of  the  fourth,  1  billion ;  and 
so,  for  periods,  still  to  the  left. 

4.  To  quadrillions  succeed  quintillions,  sextillions,  septillions,  &c. 

5.  The  pupil  should  be   required  to   commit,  thoroughly,  the 
names  of  the  periods,  so  as  to  repeat  them  in  their  regular  order 
from  left  to  right,  as  well  as  from  right  to  left. 


NOTATION   AND   NUMERATION.  19 

Rule  for  Notation. 

1st.  Begin  at  the  left  hand  and  write  each  period,  as  if 
it  were  a  period  of  units. 

2d.  When  the  number  in  any  period,  except  the  left-hand 
period,  is  expressed  by  less  than  three  figures,  prefix  one 
or  two  ciphers ;  and  when  a  vacant  period  occurs,  fill  it  with 
ciphers. 

Examples  in  Notation. 

Express  the  following  numbers  in  figures  : 

1.  One  hundred  and  five. 

2.  Three  hundred  and  two. 

3.  Five  hundred  and  nineteen. 

4.  jOne  thousand  and  four. 

5.  Eight  thousand,  seven  hundred  and  one. 

6.  Forty  thousand,  four  hundred  and  six. 

7.  Fifty-eight  thousand  and  sixty-one. 

8.  Ninety-nine  thousand,  nine  hundred  and  ninety-nine. 

9.  Four  hundred  and  six  thousand  and  forty-nine. 

10.  Six  hundred   and  forty-one   thousand,   seven   hundred 
and  twenty-one. 

11.  One  million,  four  hundred   and   twenty-one   thousand, 
six  hundred  and  two. 

12.  Nine   millions,  six  hundred   and   twenty-one  thousand 
and  sixteen. 

13.  Ninety-four  millions,  eight   hundred   and  seven   thou- 
sand, four  hundred  and  nine. 

14.  Four  billions,  three  hundred  and  six  thousand,  nine 
hundred  and  nine. 

15.  Forty-nine  trillions,  nine  hundred  and  forty-nine  thou- 
sand and  sixty-five. 

16.  Nine   hundred   and  ninety   quadrillions,   nine  hundred 
and  ninety-nine  millions,  nine  hundred  and  ninety  thousand, 
nine  hundred  and  ninety-nine. 

It.  Four  hundred  and  nine  sextillions,  two  hundred  and 
nine  thousand,  one  hundred  and  six. 


20 


NOTATION    AND    NUMERATION. 


Rule  for  Numeration. 

I.  Divide  the  number  into  periods  of  three  figures  each, 
beginning  at  the  right  hand. 

II.  Name  the  order  of  each  figure,  beginning  at  the  right 
hand. 

III.  Then,  beginning  at  the  left  hand,  read  each  period 
as  if  it  stood  alone,  naming  its  unit. 


Examples  in  Numeration. 

Let  the  pupil  point  off  and  read  the  following  numbers ; 
then  write  them  in  words : 

804321049 
90067236708 
870432697082 

16.  1704291672301 

17.  3409672103604 

18.  49701342641714 


1. 

67 

7. 

6124076 

13. 

2. 

125 

8. 

8073405 

H. 

3. 

6256 

9. 

26940123 

15. 

4. 

4697 

10. 

9602316 

16. 

5. 

23697 

11. 

87000032 

17. 

6. 

412304 

12. 

1987004086 

18. 

19.  8760218760541 

20.  904326170365 

21.  30267821040291 

22.  907620380467026 


23.  9080620359704567 

24.  9806071234560078 

25.  30621890367081263 

26.  350673123051672607 


NOTE. — Let  each  of  the  above  examples,  after  being  written  on 
the  blackboard,  be  analyzed  as  a  class  exercise ;  thus : 

Ex.  1.   How  many  tens  in  67?    How  many  units  over? 

2.  In  125,  how  many  hundreds  in  the  hundreds  place?     How 
many  tens  in  the  tens  place  ?     How  many  units  in  the  units  place  ? 
How  many  tens  in  the  number? 

3.  In  6256,  how  many  thousands  in  the  thousands  place  ?     How 
many  hundreds  in  the  hundreds  place?     How  many  tens  in  the 
tens  place  ?    How  many  units  in  the  units  place  ? 

4.  In  4697,  how  many  tens  are  there?     The  6  hundreds  are 
equal  to  how  many  tens?     To  how  many  tens  are  the  4  thou- 
sands equal?     To  how  many  tens  are  the  4  thousands  and  6 
hundreds  equal? 


NOTATION   AND   NUMERATION.  21 

Examples  in  Notation  and  Numeration 

1.  Write  two  hundred  and  nine. 

2.  Write  five  thousand  and  five. 

3.  Write  twelve  thousand  and  twelve. 

4.  Read  1040;   30706;   6606. 

5.  Read  2001 ;  35006 ;   4070070. 

6.  Write  one  hundred  thousand,  one  hundred  and  one. 

7.  Read  207600042;   1000860005. 

8.  Read  100000100;   5000000750001. 

9.  Write  forty-seven  millions,  two  hundred  and  four  thou- 
sand, eight  hundred  and  fifty-one. 

10.  Write  six  quadrillions,  forty-nine  trillions,  seventy-two 
billions,  four  hundred  and  seven  thousand,  eight  hundred  and 
sixty-one. 

11.  Write  eight  hundred  and  ninety-nine  quadrillions,  four 
hundred  and  sixty  trillions,  eight  hundred  and  fifty  billions, 
two  hundred  millions,  five   hundred   and   six   thousand,  four 
hundred  and  ninety-nine. 

12.  Write  and  read,  fifty-nine  trillions,  fifty-nine  billions, 
fifty-nine  millions,  fifty-nine  thousand,  nine  hundred  and  fifty- 
nine. 

13.  Eleven  thousand,  eleven  hundred  and  eleven. 

14.  Nine  billions  and  sixty-five. 

15.  Write  and  read,  three  hundred  and  four  trillions,  one 
million,  three  hundred  and  twenty-one  thousand,  nine  hundred 
and  forty-one. 

16.  Write  and  read,  nine  trillions,  six  hundred  and  forty 
billions,  with  7  units  of  the  ninth  order,   6  of  the  seventh 
order,  8  of  the  fifth,  2  of  the  third,  1  of  the  second,  and  3 
of  the  first. 

17.  Write  and  read,  three  hundred  and  five  trillions,  one 
hundred  and  four  billions,  one  million,  with  4  units  of  the 
fifth  order,  5  of  the  fourth,  7  of  the  second,  and  4  of  the 
first. 

18.  Write  and  read,  three  hundred  and  one  billions,  si* 
millions,  four  thousand,  with  8  units  of  the  fourteenth  order, 
6  of  the  third,  and  two  of  the  second. 


22  ADDITION   OF 


ADDITION. 

24.  1.  John  has  two  apples,  and  Charles  has  three:  how 
many  have  both  ? 

ANALYSIS. — They  have,  together,  as  many  apples  as  are  equal 
to  2  apples  counted  with  3  apples,  which  are  5  apples. 

2.  James  had  5  marbles,  and  William  gave  him  7  more, 
how  many  had  he  then  ? 

3.  Mary  has  6  pins,  and  Jane  9  :  how  many  have  both  ? 

4.  How  many  are  5  and  3  ?     6  and  4  ? 

5.  How  many  are  4  and  9  ?     8  and  5  ? 

6.  How  many  are  3  and  7  ?     10  and  0  ?     0  and  10  ? 

7.  How  many  are  1  and  5  and  6  ?     3  and  4  and  9  ? 

The  answer  to  any  of  the  above  questions,  is  called,  the  Sum 
of  the  numbers,  and  the  operation  by  which  we  find  it,  is  called, 
Addition. 

25.  The  SUM   of  two  or  more  numbers,  is  a  number  which 
contains  as  many  units  as  there  are  in  all  the  numbers  added. 

ADDITION  is  the  operation  of  finding  the  sum  of  two  or 
more  numbers. 

Of  the  Signs. 

26.  The  sign   +,   is    called  plus,   which    signifies,   more. 
When  placed  between  two  numbers,  it  denotes  that  they  are 
to  be  added  together. 

The  sign  =,  is  called,  the  sign  of  equality.  When  placed 
between  two  numbers,  it  denotes  that  they  are  equal  to  each 
other.  Thus,  3  +  2  =  5,  denotes  that  the  sum  of  3  and  2 
is  equal  to  5. 

24.  What  is  the  sum  of  two  or  more  numbers  ?  What  is  Addi 
tion? 

26.  What  is  the  sign  of  Addition  ?  What  is  it  called  ?  What 
does  it  signify  ?  Express  the  sign  of  equality.  When  placed  be- 
tween two  numbers,  what  does  it  show  ? 


B1MPLK   NUMBERS. 

Addition  Table. 


2+    0=    2 

3+    0=    3 

4+0=4 

5+0=5 

2+1=3 

3+    1=    4 

4+1=5 

5+1=6 

2+    2=    4 

3+    2=    5 

4+    2=    6 

5+    2=    7 

2+    3=    5 

3+3=6 

4+    3=    7 

5+    3=    8 

2+    4=    6 

3+    4=    7 

4+4=8 

5+4=9 

2+    5=    7 

3+    5=    8 

4+    5=    9 

5+    5  =  10 

2+    6=    8 

3+    6=    9 

4+    6=10 

5+    6  =  11 

2+    7=    9 

3+    7  =  10 

4+    7  =  11 

5+    7  =  12 

2+    8  =  10 

3+    8  =  11 

4  +    8  =  12 

5+    8  =  13 

2+    9  =  11 

3  +    9  =  12 

4  +    9=13 

5+    9  =  14 

2+10  =  12 

3  +  10  =  13 

4  +  10  =  14 

5  +  10  =  15 

6  +    0  =  '6' 

7+0=7 

8+    0=    8 

9+    0=    9 

6+    1=    7 

7+1=8 

8+    1=    9 

9+    1  =  10 

6+    2=    8 

7+2=9 

8+    2  =  10 

9+    2  =  11 

6+3=9 

7  +    3  =  10 

8+    3  =  11 

9+    3  =  12 

6+    4  =  10 

7  +    4  =  11 

8+    4  =  12 

9+    4  =  13 

6+    5  =  11 

7  +    5  =  12 

8+    5  =  13 

9+    5  =  14 

6+    6  =  12 

7  +    6  =  13 

8+    6  =  14 

9+    6  =  15 

6+    7  =  13 

7  +    7  =  14 

8+    7  =  15 

9+    7  =  16 

6+    8  =  14 

7  +    8  =  15 

8+    8  =  16 

9+    8  =  17 

6+    9  =  15 

7  +    9=  16 

8+    9=  17 

9+    9  =  18 

6  +  10  =  16 

7  +  10  =  17 

'8  +  10  =  18 

9  +  10  =  19 

2  +  3  = 

1+2+4= 

2+3+5+ 1 = 

6+7+2+3= 

1+6+7+2+3= 

1+2  +  3  +  4  + 5.+ 6  +  7  +  8  +  9  = 


how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many? 

27.   The  operation  of  Addition  is  governed  by  four  prin- 
ciples, viz.  : 

1.  A  single  number  expresses  a  collection  of  like  units. 

2.  Like  units  alone  can  be  added  together  ;  that  is,  units 
must  be  added  to  units,  tens  to  tens,  hundreds  to  hundreds, 
&c. 

3.  Every  number,  expressed  by  .two  or  more  figures,  is  the 


24  ADDITION   OF 

sum  of  its  units,  tens,  hundreds,  &c. ;   thus,  279  is  the  sum 
of  2  hundreds,  7  tens,  and  9  units 

4.    The  sum  of  several  numbers  is  equal  to  the  sum  of  all 
their  parts. 

1.  James  has  14  cents,  and  John  gives  him  21 :  how  many 
cents  has  he  then  ? 

OPERATION. 

ANALYSIS. — Since  units  must  he  added  to  units,         ... 

and  tens  to  tens,  the  numbers  are  written  so  that         <,, 
units  of  the  same  order  may  fall  in  the  same  col-        — 
umn,    and    a   line    is    drawn   beneath   them.     The         35  cents, 
column  of  the  lowest  order  is  first  added,  and  con- 
tains 5  units,  which  are  written  under  the  column.     The  tens  are 
next  added,  and  they  amount  to  3  tens,  which  are  written  under 
the  tens.     The  sum  is  3  tens  and  5  units,  or  thirty-five. 

2.  A  gentleman  bought  a  carriage  for  385  dollars,  -a  team 
of  horses   for  286  dollars,  and  two   sets   of  harness   for  96 
dollars  :   what  did  he  pay  for  all  ?  OPERATION. 

ANALYSIS. — Write  the  numbers  so  that  units  of  the  385 
same  value  shall  fall  in  the  same  column;  then  add  286 
each  order  of  units  separately.  96 

Sum  of  the  units 17 

Sum  of  the  tens 25 

Sum  of  the  hundreds 5 

Sura  total 767 

The  following,  however,  is  the  method  in  practice: 

ANALYSIS. — Write  the  numbers  as  before.  The 
units  are  added  together,  and  their  sum  is  17,    OPERATION- 
which  is  1  ten  and  7  units;  the  units  are  placed          385 
under  the   column  of  units,  and  the  1'ten  is 
added  with   the    column    of  tens,   which   then 
amounts  to  26  tens,  equal  to  2  hundreds  and  6          >j67  dollars, 
tens;  the  6  tens  are  placed  under  the  tens,  and 
the  hundreds   are   added  with  the   column  of  hundreds,  which 
amounts  to  7,  and  is  therefore  placed  under  the  hundreds. 

27.  By  how  many  principles  is  the  operation  of  adding  governed 
Name  them. 


SIMI'LK    NUMBKKS.  25 

When  a  column  amounts  to  ten,  or  more  than  ten,  the  nnit  figure 
IB  set  down,  and  the  tens'  figure  is  added  to  the  next  column,  be- 
cause, 10  units  of  any  order  make  1  unit  of  the  next  higher  order. 
This  process  is  called,  carrying  to  the  next  column. 

*  28.   Hence,  to  find  the  sum  of  two  or  more  numbers,  we 
have  the  following 

Rule. 

I.  Write  the  numbers  to  be  added,  so  that  units  of  thr 
same  order  shall  fall  in  the  same  column. 

II.  Add  the  column  of  units :  set  down  the  units  of  the 
sum,  and  carry  the  tens  to  the  next  column. 

HI.  Add  the  column  of  tens :  set  down  the  tens,  and 
carry  the  hundreds  to  the  next  column  ;  and  so  on,  till  all 
the  columns  are  added,  and  set  down  the  entire  sum  of  the 
last  column. 

Proof. 

The  PROOF  of  any  operation  in  Addition,  consists  in  show- 
ing that  the  result,  or  answer,  contains  as  many  units  as  there 
are  in  all  the  numbers  added,  and  no  more.  There  are  two 
methods  of  proof,  for  beginners: 

I.  Begin  at  the  top  of  the  units  column,  and  add  all  the 
columns  downward,  carrying  from  one  column  to  the  other, 
as  when  they  were  added  upward.     If  the  two  results  agree, 
the  work  is  supposed  to  be  right. 

II.  Draw  a  line,  dividing  the  numbers  into  parts.     Add 
the  parts  separately,  and  then  acid  the  sums.     If  the  last 
sum  is  the  same  as  the  sum  Jirst  found,  the  work  may  be 
regarded  as  right. 

28.  How  do  you  set  down  numbers  for  addition  ?  Where  do  you 
begin  to  add?  If  the  sum  of  any  column  can  be  expressed  by  a 
single  figure,  what  do  you  do  with  it  ?  When  it  cannot,  what  do 
you  do?  When  you  add  to  the  next  column,  what  is  it  called? 
What  do  you  set  down  in  the  last  column  ?  What  docs  the  proof 
consist  of,  in  Addition?  What  is  the  first  method  of  proof?  What 
is  the  second  method  of  proof? 


26  ADDITION    OF 

Reading. 

The  pupil  should  be  early  taught  to  omit  the  intermediate 
words  in  the  addition  of  a  column  of  figures.  Thus,  in  example 
18,  instead  of  saying,  7  and  5  are  12  and  1  are  13  and  6  are  10; 
he  should  say,  twelve,  thirteen,  nineteen;  and  in  the  column  of 
tens,  ten,  nineteen,  twenty-three ;  and  similarly  for  the  other 
columns.  This  is  called,  reading  the  columns.  Let  the  pupils  he 
often  practiced  in  it,  both  separately,  and  in  concert  in  classes. 

Examples. 

1.  A  farmer  has  160  sheep  in  one   field,  20  in  another, 
and  16  in  another :  how  many  has  he  in  all  ? 

2.  If  a  gentleman  travels   328  miles  one  day,  171  miles 
the  next  day,  and  250  miles  the  third  day,  how  far  will  he 
travel  in  all? 

(3.)  (4.)  (5.)  (6.) 

604  199  776  398 

743  367  407  999 

(7.)  (8.)  (9.)  (10.) 

427  329  3034  8094 

242  260  6525  1602 

330  100  230  103 

(11.)  (12.)  (13.)  (14.) 

4096  9976  9875  67954 

3271  8757  9988  98765 

4722  8168  8774  37214 

(15.)  (16.)  (17.)  (18.) 

6412  90467  87032  432046 

1091  10418  64108  210491 

6741  91467  74981  809765 

9028  41290  21360  542137 


SIMPLE    NUMBERS. 


(19.) 

21467 

80491 

67421 

4304 

2191 


(20.) 

89479 

75416 

7647 

214 

19 


(21.) 

74167 

21094 

2947 

674 

85 


(22.) 

9947621 

704126 

81267 

9241 

495 


Proof  of  Addition. 

Either  of  the  following  methods  may  be  used  in  proving 
examples  in  Addition. 

(23.) 


riioop. 


34578  I      84578 


3750^ 
87 

328  [ 
17 

327  J 

Sum,  39087 

(25.)        j 
672981043  *- 
67126459    ' 
39412767 
7891234 
109126 
84172 
72120 


4509 


(24.) 
23456) 
*-  78901  [ 
23456 ) 

78901 ) 
23456  5- 
78901  \ 


PliOOF. 

1^5813 


39087    Sum,  3Q7071    307071 


(26.) 

>9L278976 

7654301 

876120 

723456 

31309 

4871 

978 


.X$416785413 

6915123460 

31810213 

7367985 

654321 

47853 

2685 


28.  What  is  the  sum  of  304  and  273  ? 

£  29.  What  is  the  sum  of  3607  and  4082  ? 

|  30.  What  is  the  sum  of  30704  and  471912? 

^,31.  What  is  the  sum  of  398463  and  401536? 

32.  If  a  top  costs  6  cents,  a  knife  25  cents,  a  slate  12 
cents :  what  does  the  whole  amount  to  ? 

33.  John  gave  30  cents  for  a  bunch  of  quills,  18  cents 
for  an  inkstand,  25  cents  for  a  quire  of  paper :  what  did  the 
wholo  cost  him  ?   >j 


28  ADDITION    OF 

I 

34.  If  2  cows  cost  143  dollars,  5  horses  621  dollars,  and 

2  yoke  of  oxen  124  dollars,  what  will  be  the  cost  of  them  all  ? 

35.  What  is  the  sum  of  8  hundreds,  4  tens,  6  units,  and 
6  thousands  ? 

36.  What  is  the  sum  of  3  units,  5  units,  6  tens,  3  tens, 
4  hundreds,  3  hundreds,  %5  thousands,  and  4  thousands  ? 

37.  What  is  the  sum -of  five  units  of  the  4th  ordes,  1  of 
the  3d,  three*  of  the  4th,  five  of  the  3d,  and  one  of  th\lst? 

38.  What  is  the  sum  of  six  units  of  the  2d  order,  five  of 
the  3d,  six  of  the  4th,  three  of  the  2d,  four  of  the  3d,  two 
of  the  1st,  and  four  of  the  2d  ? 

_  39.   What  is  the  sum  of  3  and  6,  5  tens  and  2  tens,  and 

3  hundreds  and  6  hundreds  ? 

40.   What  is  the  sum  of  4  and  5,  5  tens,  3  hundreds  and 
2  hundreds  ?  « 

y   41.    Add   8635,    2194,    7421,    5063,    2196,  and  1245   to- 
gether. 

y       42.    Add   246034,   298765,   47321,    58653,    64218,    5376, 
9821,  and  340  together.    ^ 

f'      43.    Add   27104,    32547,   10758,    6256,   704321,   730491. 
*  2787316,  and  2749104  together.      «» 

44.    Add  1,  37,  39504,  6890312,   18757421,  and  265  to, 
jther.          «* 

"^^  45.  What  is  the  sum  of  the  following  numbers,  viz. : 
seventy-five  ;  one  thousand  and  ninety-five ;  six  thousand  four 
hundred  and  thirty-five ;  two  hundred  and  sixty-seven  thou- 
sand ;  one  thousand  four  hundred  and  fifty-five ;  twenty-seven 
millions  and  eighteen;  two  hundred  and  seventy  millions  and 
twenty-seven  thousand  ? 

46.  What    is    the    sum    of    372856,    404932,    2704793, 
9078961,  304165,  207708,  41274,  375,  271,  34,  and  6? 

47.  What   is   the   sum   of    4073678,    4084162,    3714567, 
27413121,  27049,  87419,  27413,  604,  37,  and  9? 

48.  What  is  the  sum  of  36704321,  2947603,  999987,  76, 
47213694,  21612090,  8746,  31210496,  and  3021 ?    ! 


SIMPLE   NUMBERS. 


29 


49.  Add  together  fifty-eight  billions,  nine  hundred  and 
eighty-two  millions,  four  hundred  and  eighty-seven  thousand, 
six  hundred  and  fifty-four ;  seven  hundred  and  forty  billions, 
three  hundred  and  fifty  millions,  five  hundred  and  forty 
thousand,  seven  hundred  and  sixty ;  four  hundred  and  twenty- 
five  billions,  seven  hundred  and  three  millions,  four  hundred 
and  two  thousand,  six  hundred  and  three ;  thirty-four  bil- 
lions, twenty  millions,  forty  thousand  and  twenty;  five  hun- 
dred and  sixty  billions,  eight  hundred  millions,  seven  hundred 
thousand  and  five  hundred. 


(50.) 

87406 
89507 
41299 
47208 
71G25 
72428 
97206 
41278 
28907 
25412 
27049 
28416 
72204 
70412 
27426 
62081 
81697 
87489 
21642 
24672 


(51.) 

92674 
27049 
28372 
37041 
49741 
57214 
59261 
41219 
57267 
40216 
87614 
92742 
87046 
90212 
17618 
40261 
57274 
21859 
42673 
51814 


(52.) 

25043 
97069 
81216 
75850 
90417 
19216 

20428 

60594 

72859 

o  43706 

g  21441 

87604 

71215 


18972 
27042 
59876 
54301 
87415 
32018 
72687 


388811 


»  377847 


352311 


53.  By  the  census  of  1850,  the  population  of  the  ten  largest 
cities  was  as  follows :  New  York,  515547 ;  Philadelphia, 
340045;  Baltimore,  169054;  Boston,  136881  ;  New  Orleans, 
116375;  Cincinnati,  115436;  Brooklyn,  96838;  St.  Louis, 
77860  ;  Albany,  50763  ;  Pittsburgh,  46601 :  what  was  their 
entire  population  ? 


30  ADDITION   OF 

Applications. 

29.  In  all  the  applications  of  Arithmetic,  the  numbers 
added  together  must  have  the  same  unit. 

In  the  question,  How  many  head  of  live  stock  in  a  field, 
there  being  6  cows,  2  oxen,  3  steers,  and  15  sheep,  the  unit 
is  1  head  of  live  stock.  And  the  same  principle  is  applica- 
ble to  all  similar  questions. 

1.  How   many   days    are    there    in    the    twelve   calendar 
months  ?     January  has  31,  February  28,  March  31,  April  30, 
May  31,  June  30,  July  31,  August  31,  September  30,  Oc- 
tober 31,  November  30,  and  December  31. 

2.  A   merchant   bought   a   horse   for    1JL2   dollars  ;    after 
keeping  him  a  short  time,  he   sold  him,  and  gained  25  dol- 
lars :  how  much  did  he  receive  for  the  horse  ? 

;3.    A  person  sold  a  house  and  lot  for  3650  dollars,  and, 
so  doing,  lost  375  dollars :  what  had  they  cost  him  ? 

4.  A  speculator  bought  a  house  and  lot  for  1964  dollars, 
expended  384  dollars  in  repairing  and  refitting  the  property, 
paid  taxes  and  insurance  amounting  to  56  dollars,  and  then 
sold   so   as   to   gain  396  dollars :   what  did  he  get  for  the 
property  ? 

5.  What  is  the  total  weight  of  seven  casks  of  merchan- 
dise ;  No.  1  weighing  960  pounds,  No.  2,  725  pounds,  No.  3, 
830  pounds,  No.  4,  798  pounds,  No.  5,  698  pounds,  No.  6, 
569  pounds,  No.  7,  987  pounds  ? 

6.  At  the  Custom  House,  on  the  first  day  of  June,  there 
were  entered  1800  yards  of  linen ;  on  the  10th,  2500  yards  ; 
on  the  25th,  600  yards  ;  on  the  day  following^  7500  yards ; 
and  on  the  last  three  days  of  the  month,  1325  yards  each  day : 
what  was  the  whole  amount  entered  during  the  month  ? 

7.  A  farmer  has  his  live-stock  distributed  in  the  following 
mainier  :   in  pasture  No.  1  there   are  5  horses,  14  cows,  8 

29.  What  principles  govern  all  the  additions  in  Arithmetic? 
What  is  the  unit  in  the  question,  How  many  head  of  cattle  in  a 
pasture  ? 


SIMPLE   NUMBERS.  31 

oxen,  and  6  colts  ;  in  pasture  No.  2,  8  horses,  4  colts,  6 
cows,  20  calves,  and  12  head  of  young  cattle  ;  in  pasture 
No.  3,  320  sheep,  16  calves,  two  colts,  and  5  head  of  young 
cattle.  How  much  live-stock  had  he  of  each  kind,  and  how 
many  head  had  he  altogether  ? 

8.  What  is  the  interval  of  time  between  an  event  which 
happened   125   years   ago,    and    one   that  will  happen   267 
years  hence  ? 

9.  In  1861,  John  was  24  years  old:  in  what  year,  should 
he  live,  will  he  be  1 9  years  older  ? 

10.  A  merchant  has  real  estate  worth  45615  dollars,  fur- 
niture worth  2862  dollars,  merchandise  in  store  worth  25659 
dollars,  cash  in  bank  9645  dollars,   and  cash  on  hand  456 
dollars :  what  is  the  amount  of  his  fortune  ? 

11.  Of  a  debt,  George  paid  475  dollars,  and  Samuel  paid 
the  remainder,  which  was  625  dollars  :  what  was  the  debt  ? 

12.  A  lot  of  ground  cost  750  dollars ;  a  house  was  erected 
thereon,  whose  cost  was,  for  carpenters'  work,  2000  dollars, 
masons'  work,  765  dollars,  painters'  work,  265  dollars,  and 
for  other  work,   327   dollars  :   what  was   the  cost   of  house 
and  lot  ? 

13.  There  are  60  seconds  in  a  minute,  3600  in  an  hour, 
86400  in  a  day,   604800   in  a  week,   2419200  in  a  month, 
and  31557600  in  a  common  year:  how  many  seconds  in  the 
periods  of  time  named  above  ? 

14.  Suppose  a  merchant  to  buy  the  following  parcels  of 
cloth:    3912   yards,    1856,    2011,    4540,    937,    6338,    3603, 
1586,  2044,  2951,  4228,  1345,  1011,  6138,  960,  607,  5150, 
43886,   617,   7513,   4079,   743,   612,   2519,  1238,  and  2445 
yards :  how  many  yards  does  he  buy  in  all  ? 

15.  What  is  the  sum  of  two  millions  bushels  of  corn,  five 
hundred   and  thirty-one  thousand  bushels,   one   hundred   and 
twenty  bushels,    fourteen   thousand   bushels,   thirty  thousand 
and  twenty-four  bushels,  five  hundred  and  sixty  bushels,  and 
seven  hundred  and  two  bushels  ? 

16.  The  mail  route  from  Albany  to  New  York  is  144  miles, 
from  New  York  to  Philadelphia  90  miles,  from  Philadelphia 


32  ADDITION. 

to  Baltimore  98  miles,  and  from  Baltimore  to  Washington  City 
38  miles :  what  is  the  distance  from  Albany  to  Washington  ? 

17.  A  man,  dying,  leaves  to  his  only  daughter  nine  hun- 
dred and  ninety-nine  dollars,  and  to  each  of  three  sons  two 
hundred  dollars  more  than  he  left  the  daughter:   what  was 
each  son's  portion,  and  what  the  amount  of  the  whole  estate  ? 

18.  The  number  of  acres  of  the  public  lands  sold  in  1834 
was    4658218  ;    in    1835,    12564478 ;    in    1836,    25167833. 
The  number  of  acres  sold  in  1840  was  2236889;   in  1841, 
1164796;  in  1842,  1129217.     How  many  acres  were  sold-in 
the  first  three,  and  how  many  in  the  last  three  years  ? 

19.  What  was  the  population  of  the  British  provinces  in 
North  America  in  1834,  the   population  of  Lower  Canada 
being  stated  at  549005  ;  of  Upper  Canada,  336461 ;  of  New 
Brunswick,  152156  ;  of  Nova  Scotia  and  Cape  Breton,  142548  ; 
of  Prince  Edward's  Island,  32292 ;  of  Newfoundland,  75000  ? 

20.  By  the  census  of*  1850,  the  number  of  deaf  and  dumb 
in  the  United  States  was   9803  ;  of  blind,  9794 ;  of  insane, 
15610 ;  of  idiots,  15787  :  what  was  the  aggregate  ? 

21.  A  vessel  took  for  her  cargo,  from  New  York  to  Lon- 
don, cotton,  wheat,  flour,  corn,  and  tobacco  ;  the  cotton  was 
valued  at  16562  dollars,  the  wheat  5690  dollars,   the  flour 
25645  dollars,  the  corn  10684  dollars,  and  the  tobacco  35760 
dollars :  what  was  the  value  of  the  cargo  ? 

22.  By  the  census  of  1850,  the  population  of  the  District 
of   Columbia  was    51687  ;    of  the   Territory   of  Minnesota, 
6077  ;  of  New  Mexico,  61547  ;  of  Oregon,  13294  ;  of  Utah, 
11380  :  what  was  the  population  of  the  Territories,  including 
the  District  of  Columbia  ? 

23.  By  the  census  of  1850,  the  population  of  Maine  was 
583169;  of  New  Hampshire,  317976;  of  Vermont,  314120; 
of  Massachusetts,  994514  ;    of  Rhode  Island,  147545  ;    and 
of  Connecticut,  370792  :  what  was  the  population  of  the  six 
New  England  States? 

24.  A  person,  who  was  born  in  1801,  died  at  the  age  of 
46  years  :  his  son  died  15  years  afterward :  in  what  year  did 
the  son  die  ? 


SUBTRACTION.  33 


SUBTRACTION. 

30.  John;  has  5  apples,  and  Charles  2  :    how  many  more 
apples  has  John  than  Charles  ? 

ANALYSIS. — As  many  more  as,  added  to  what  Charles  has,  will 
make  his  number  equal  to  John's :  3  added  to  2,  gives  5 :  There- 
fore, John  has  3  apples  more  than  Charles. 

31.  The  DIFFERENCE    between  two  numbers,  is  that  num- 
ber which  added  to  the  less,  will  give  the  greater. 

32.  SUBTRACTION   is  the  operation  of  finding  the  difference 
between  two  numbers. 

33.  The  MINUEND   is  the  greater  of  the  two  numbers. 

34.  The  SUBTRAHEND    is  the  less  of  the  two  numbers. 

35.  The  REMAINDER    or  Difference,  is   the  result  of  the 
operation. 

If  the  numbers  are  equal,  the  remainder  is  0,  whichever  be 
taken  as  the  minuend. 

Of  the  Signs. 

36.  The   sign  — ,   is   called   minus,   which   signifies,    less. 
When  placed  between  two  numbers,  it  denotes  that  the  one 
before  it  is  the  minuend,  and  the  one  after  it,  the  subtra- 
hend  ;  thus,  5-3  =  2, 

denotes  that  5  is  the  minuend,  3  the  subtrahend,  and  2  the 
remainder. 


8  —  4  =     how  many  ? 
12  —  5  =     how  many  ? 


17  —    8  =     how  many? 
19  -  10  =     how  many? 


37.   The  principles  which  control  the   operations   of  Sub- 
traction are, 

1.  That  the  difference,  added  to   the  less   number,  gives 
the  greater. 

2.  That  the  minuend  and  subtrahend  must  have  the  same 
unit. 

2* 


34  SUBTRACTION   OF 

38.    1.  James  has  27  apples,  and  gives  14  to  John :  how 
many  has  he  left  ? 

ANALYSIS. — The  27  is  made  up  of  7  units    OPERATION. 
and  2  tens;  and  the  14,  of  4  units  and  1  ten.          27  Minuend. 
Subtract  4  units  from  7  units,   and  3  units          14  Subtrahend, 
will  remain;  subtract  1  ten  from  2  tens,  and 
1  ten  will  remain:   hence,  the  remainder  is         15  Remainder- 
13. 

2.  A  farmer  had  378  sheep,  and  sold  256 :  how  many  had 

he  left  ? 

ANALYSIS. — We  first  write  the  number  378,  and  OPERATION. 
then  256  under  it,  so  that  units  of  the  same  order  378 

shall  fall  in  the  same  column.     We  then  take  6  units  256 

from  the  8  units,  5  tens  from  7  tens,  and  2  hundreds  _ 

from  3  hundreds,  leaving  for  the  remainder,  122. 

3.  What  is  the  difference  between  843  and  562  ? 

ANALYSIS. — Begin  at  the  units'  column,  and  sub- 
tract 2  from  3.    At  the  next  place  we  meet  a  diffi-  OPERATION 
culty,  for  we  can  not  subtract  a  greater  number  from  843 
a  less.  562 

If  we  take   1   from    the    8  hundreds    (equal   to 

"10  tens),  and  add  it  to  the  4  tens,  the  minuend  will  714     3 

become  7  hundreds,  14  tens,  and  3  units.     We  then  5     g     2 

say,  6  tens  from  14  tens  leaves  8  tens ;  and  5  hun-  

dreds  from  7  hundreds  leaves  2  hundreds:  hence,  "     ° i 

the  remainder  is  281. 

The  same  result  is  obtained  by  adding,  mentally,  10  10 

to  the  4  tens,  and  then  adding  1  to  5,  the  next  figure  843 

of  the  subtrahend  at  the  left;  for,  adding  1  to  the  5  is  562 

the  same  as  diminishing  the  8  by  1.     This  process  of  1 

adding  10  to  a  figure  of  the  minuend,  and  1  to  the  next  281 
figure  of  the  subtrahend,  at  the  left,  is  called,  borrowing. 


81.  What  is  the  difference  between  two  numbers? — 32.  What  is 
Subtraction  ? — 33.  What  is  the  minuend  ?— 34.  What  is  the  subtra 
hend?— 35.  What  is  the  remainder  ?— 36.  What  is  the  sign  of 
Subtraction  ? — 37.  What  are  the  principles  which  control  the  oper- 
ations of  Subtraction  ? 


SIMPLE   NUMBERS.  35 

Hence,  for  Subtraction,  we  have  the  following 

Rule. 

I.  Write  the  less  number  under  the  greater,  so  tliat  units 
of  the  same  order  shall  fall  in  the  same  column. 

II.  Begfa  at  the  right  hand,  and  subtract  each  figure  of 
the  subtrahend  from  the  one  directly  over  it,  when  the  upper 
figure  is  the  greater. 

III.  When  the  upper  figure  is  the  less,  add  10  to  it,  before 
subtracting,  and  then  add  1  to  the  next  figure  of  the  sub- 
trahend. 

Proof. 

i 

Add  the   remainder  to   the  subtrahend.     If  the  work  is 
right,  the  sum  will  be  equal  to  the  minuend. 

Spelling — Reading. 

39.  What  is  the  difference  between  725  and  341  ? 

By  the  common  method,  which  is  spelling,  we  say, 

1  from  5  leaves  4;  4  from  12  leaves  8;  1  to  carry  OPEBATION. 
to  3  are  4;  4  from  V  leaves  3.  725 

Reading  the  words  which  express  the  final  result,          341 
we  should  make  the  operations  mentally,  and  say,  OQ . 

A      ft      Q  •  6°* 

4,  8,  3. 

Let  the  pupils  be  practiced  separately  in  the  reading,  and  also 
in  concert  in  classes. 

Examples. 


(1.) 

(2.) 

(3.) 

(4.) 

Minuends,       874 

999 

8497 

62843 

Subtrahends,  642 

367 

7487 

51720 

Remainders,    232 

38.  Give  the  rule  for  Subtraction. 

89.   Explain  what  is  meant  by  spelling  and 


36 


SUBTRACTION   OF 


(5.) 
972 
631 


(6.) 
278846 
167504 


894862 
170641 


(8.) 

27609465 
17206105 


(9.) 
32780 
15678 


(10.) 

8592678 
1078953 


(11.) 

67942139 
9756783 


219067803 
104202196 


(13.) 
4760 
1986 

(18.) 

30000 

9999 


(14.) 
3600 
1863  AV 

(19.) 
100000 


(15.) 

87000 
1009 


(16.) 

67087 
40000 


(20.) 
20006400 
18609635 


(wo 

10000 


(21.) 

100044400 
90094406 


j/22.    From  2637804  take  2376982. 
I  23.    From  3^62162  take  82654l>~ 

24.  From  78213609  take  27821890.  x 

25.  From    thirty    thousand    and    ninety-seven,     take   one 
thousand    six  hundred  and  fifty-four,  x 

26.  From  one   hundred   millions /two  hundred  and  forty- 
seven  thousand,    take  one  million,  four  hundred  and  nine.y^ 

,27.    Subtract  one  from  one  million.  ^ 
28.    From  804367  subtract  2t?05.  ^ 
i  29.    From  18623041  subtract  6f¥94.  ^ 

30.  From  4270492  subtract  26409. 

31.  From  8741209  subtract  728.104. 

32.  From  741874  subtract  689346. 

33.  From  sixty-five  billions,  three  millions,  six  hundred  and 
twelve,    take  nine  billions,  one  million,  four  thousand  and  6. 

34.  From  14  billions,  127  millions,  six  hundred  and  four- 
teen thousand,  916,    take  twenty-nine  millions,  416  thousand. 

35.  From  forty  trillions,  160  billions,  42  millions,  16  thou- 
sand, f4  hundred,    take  3  trillions,  63  billions,  |l  hundred. 


SIMPLE  NUMBERS.  37 

Applications. 

40.  It  should  be  observed,  that  in  all  the  applications  of 
Subtraction,  one  number  can  be  subtracted  from  another, 
only,  when  they  both  have  the  same  unit. 

1.  Suppose  I  had  lent  a  man  1565  dollars,  and  he  died, 
owing  me  450  dollars  :  how  much  had  he  paid  me  ? 

2.  A  man  bought  a  house  and  lot  for  5650  dollars,  on  a 
mortgage  of  3  years  ;  he  paid,  at  different  times,  sums  amount- 
ing to  3756  dollars :  how  much  did  he  still  owe  ? 

3.  Suppose  John  were  born  in  eighteen  hundred  and  fifteen, 
and  James  in  eighteen  hundred  and  twenty-five :  what  is  the 
difference  of  their  ages  ? 

4.  A  man  was  born  in  1785 :  what  was  his  age  in  1830  ? 

5.  George  Washington  was  born  in  the  year  1732,  and 
died  in  1799:  how  old  was  he  at  the  time  of  his  death? 

6.  The  Declaration  of  Independence  was  published,  July 
4th,  1776:  how  many  years  to  July  4th,  1838? 

7.  In    1850,    there    wer*e    in    the    State    of    New  York, 
3,097,394   inhabitants,    and   in    the    State    of   Pennsylvania, 
2,311,786    inhabitants :    how   many   more    inhabitants    were 
there  in  New  York  than  in  Pennsylvania  ? 

8.  The  revolutionary  war  began  in  1775 ;  the  late  war,  in 
1812  :  what  time  elapsed  between  their  commencements  ? 

9.  A  merchant   bought   a  vessel   for  12642   dollars,    and 
gave  in  part  payment  a  house  that  was  worth  7585  dollars, 
and  the  rest  in  cash :  how  much  did  he  pay  in  cash  ? 

10.  A  person  sold  a  farm  for  15896  dollars,  which  had 
cost  him  12264  dollars :  how  much  did  he  gain  ? 

11.  A  man  dies,  worth   1200  dollars  ;   he  leaves  504  to 
his  daughter,  and  the  remainder  to  his  sou :  what  was  the 
son's  portion  ? 

12.  A  merchant  bought  a  house  for  6450  dollars ;  he  paid 
4625  dollars  in  cash,  and  the  rest  in  merchandise :  what  was 
the  value  of  the  merchandise  ? 

13.  Washington  died  in  1799,  at  the  age  of  67:  in  what 
year  was  he  born  ? 


38  SUBTRACTION   OF 

14.  Henry  Hudson  sailed  up  the  Hudson  river  in  1609  • 
how  many  years  since  ? 

15.  Pliny,  the  historian,  died  17  years  after  the  birth  of 
Christ :  how  many  years  was  that  before  the  Declaration  of 
Independence  ? 

16.  In  one  week,  one  steamer  traversed  2065  miles,  while 
another  traversed  1986  miles :  how  much  further  did  the  one 
travel  than  the  other  ? 

17.  In  1850,  there  were  in  New  York,  which  is  the  largest 
city  in  the  United  States,  515,547  inhabitants,  and  in  Phila- 
delphia,   the   next   largest   city,    340,045  :    how  many  more 
inhabitants  were  there  in  New  York  than  in  Philadelphia  ? 

18.  At  a  certain  period,  there  were  4338472  children  in 
the  United  States,  between  the  ages  of  5  and  15 ;   of  this 
number,  2477667  were  in  schools  :   how  many  were  out  of 
schools  ? 

19.  The  circulation  of  the  blood  was  discovered  in  1616 : 
how  many  years  to  1855  ? 

20.  A  merchant  bought  500  barrels  of  flour,  for  3500  dol- 
lars ;  he  sold  250  barrels,  for  2000  dollars  :  how  many  bar- 
rels remained  on  hand,  and  how  much  must  he  sell  them  for, 
that  he  may  lose  nothing  ? 

21.  A  merchant  bought  1675  yards  of  cloth,  for  \^Jiich  he 
paid  5025  dollars  ;  he  then  sold  335  yards,  for  1005  dollars: 
how  much  had  he  left,  and  what  did  it  cost  him  ? 

22.  In  1850,  the  slaves  in  the  United  States  amounted  to 
3204313  ;  free  colored,  to  434495  :  what  was  their  difference  ? 

23.  What  length  of  time  elapsed  between  the  birth  of  Sir 
Francis  Bacon,  in  1561,  and  the  birth  of  Benjamin  Franklin, 
in  1706? 

24.  A  merchant  sold  a  vessel  for  9768  dollars,  and,  by  so 
doing,  gained  1862  dollars:  how  much  had  the  vessel  cost? 

25.  A  householder  sold  two  houses :   for  the  first,  which 
cost  3500  dollars,  he  received  4760  dollars;  for  the  second, 
which  cost  3735  dollars,  he  received  5000  dollars :  on  which 
of  the  houses  did  he  make  the  greater  gain,  and  how  much  ? 

26.  By  the  census  of  1850,  the  number  of  white  inhabitants 


SIMPLE  NUMBERS.  39 

in  the  United  States  amounted  to  19553068  ;  and  the  blacks, 
to  3638808:  by  how  many  did  the  white  inhabitants  exceed 
the  black  ? 

27.  By  the  census  of  1850,  the  entire  population  of  the 
United  States  was  23191876;  that  of  the  six  New  England 
States,    2728116:    by  how  many  did   the  whole   population 
exceed  that  of  the  six  New  England  States  ? 

28.  An  army  of  75425  men  is  required  by  a  general,  who 
finds  that  he  has  only  49846  men :  how  many  more  men  are 
required  ? 

29.  A    boy,    in   working    an    example,    used    the    number 
2306400,  which  he  afterward  found  was  too  large  by  29875 : 
what  was  the  correct  number? 

Applications  in  Addition  and  Subtraction. 

1.  A  merchant  buys  19576  yards  of  cloth  of  one  person, 
27580  yards  of  another,  and  375  of  a  third ;   he  sells  1050 
yards  to   one  customer,   6974  yards  to  another,  and  10462 
yards  to  a  third :  how  many  yards  has  he  left  ? 

2.  A  person  borrowed  of  his  neighbor,  at  one  time,  355 
dollars,  at  another  time,  637  dollars,  and  403  dollars  at  an- 
other time  ;  he  then  paid  him,  977  dollars  :  how  much  did  he 
owe  him  ? 

3.  I  have  a  fortune  of  2543  dollars,  to  divide  among  my 
four  sons,  James,  John,  Henry,  and  Charles.     I  give  James 
504   dollars,  John  600  dollars,   and  Henry  725 :   how  much 
remains  for  Charles  ? 

4.  I  have  a  yearly  income  of  ten  thousand  dollars.     I  pay 
275  dollars  for  rent,  220  dollars  for  fuel,  35  dollars  to  the 
doctor,    and   3675   dollars   for  all  my  other  expenses :   how 
much  have  I  left  at  the  end  of  the  year  ? 

5.  A  man  pays  300  dollars  for  100  sheep,  95  dollars  for 
a  pair  of  oxen,  60  dollars  for  a  horse,  and  125  dollars  for 
a  chaise.     He  gives  100  bushels  of  wheat,  worth  125  dollars  ; 
a  cow,  worth  25  dollars ;  a  colt,  worth  40  dollars,  and  pays 
the  resfe  in  cash :  how  much  money  does  he  pay  ? 

6.  A  merchant  owes  450120  dollars,  and  has  property  as 


40  SUBTRACTION. 

follows :  bank  stock,  350000  dollars ;  western  lands,  valued 
at  225100  ;  furniture,  worth  4000  dollars,  and  a  store  of 
goods,  worth  96000 :  how  much  is  he  worth  ? 

7.  If  I  buy  489  oranges  for  912  cents,  and  sell  125  for 
186  cents,  and  then  sell  134  for  199  cents,  how  many  will 
be  left,  and  how  much  will  they  have  cost  me  ? 

8.  By  the  census  of  1850,   the   entire   population  of  the 
United  States  was  23191876  ;  the  slave  population,  3204313  ; 
free  colored,  434495 :   what  was  the  white  population  ? 

9.  A  man  gains  367   dollars,   then   loses   423 ;   a   second 
time  he  gains  875,  and  loses  912;  he  then  gains  1012  dol- 
lars :  how  much  has  he  gained  in  all  ? 

10.  If  I  agree  to  pay  a  man  36  dollars  for  plowing  25 
acres  of  land,  200  dollars  for  fencing  it,  and  150  for  culti- 
vating it,  how  much  shall  I  owe  him  after  paying  331  dollars  ? 

11.  A  merchant  bought  85  hogsheads  of  sugar  for  28675 
dollars,  paid  1231  dollars  freight,  and  then  sold  it  for  1683 
dollars  less  than  it  cost  him :  how  much  did  he  receive  for  it  ? 

12.  If  a  man's  income  is  3467  dollars  a  year,  and  he  spends 
269  dollars  for  clothing,  467  for  house  rent,  879  for  provi- 
sion, and  146  for  traveling,  how  much  will  he  have  left  at 
the  end  of  the  year  ? 

13.  Six  men  bought  a  tract  of  land,  for  36420  dollars  : 
the  first  man  paid   12140;   the  second,   3035  less  than  the 
first ;  the  third,  346 ;  the  fourth,  6070  more  than  the  third ; 
the  fifth,  1821  less  than  the  fourth :  how  much  did  the  sixth 
man  pay  ? 

14.  The   coinage   in   the    United    States    Mint,    from   its 
establishment   in  the  year  1792   to    1836,  was   thus :    gold, 
22102035  dollars;  silver,  46739182  dollars;  copper,  740331 
dollars.     The  amount  coined,  from  the  year  1837  to  1848, 
was  81436165  dollars :   how  much  more  was  corned  in  the 
last-mentioned  period  than  in  the  first  ? 


MULTIPLICATION.  41 

MULTIPLICATION. 

41.  1.  What  will  4  oranges  cost,  at  2  cents  apiece? 

ANALYSIS. — 1  orange  costs,  2  cents; 

2  oranges  cost,  2  +  2  =  4  cents ; 

3  oranges  cost,  2  +  2  +  2  =  6  cents ; 

4  oranges  cost,  2  +  2  +  2  +  2  =  8  cents. 

For  the  cost  of  1  orange,  2  is  taken  once ;  for  the  cost  of  2 
oranges,  it  is  taken  twice;  for  the  cost  of  3,  it  is  taken  three 
times ;  and  for  the  cost  of  4,  it  is  taken  four  times :  Hence,  one 
time  2  is  2;  two  times  2  are  4;  three  times  2  are  6;  and  four 
times  2  are  8. 

2.  What  is  the  cost  of  6  yards  of  ribbon,  at  7  cents  a 
yard  ? 

ANALYSIS. — Six  yards  of  ribbon  will  cost  6  times  as  much  as  1 
yard ;  since  1  yard  costs  7  cents,  6  yards  will  cost  6  times  7 
cents,  which  are  42  cents:  Therefore,  6  yards  of  ribbon,  at  7 
cents  a  yard,  will  cost  42  cents. 

42.  MULTIPLICATION  is  the  operation  of  taking  one  number 
as  many  times  as  there  are  units  in  another. 

43.  The  MULTIPLICAND    is  the  number  to  be  taken. 

44.  The   MULTIPLIER    is   the  number  denoting  how  many 
times  the  multiplicand  is  to  be  taken. 

45.  The  PRODUCT   is  the  result  of  the  operation. 

46.  The  FACTORS  of  the  product  are  the  multiplicand  and 
multiplier. 

47.  The   sign    x,   is    called  the    sign   of  Multiplication. 
When  placed  between  two  numbers,  it  denotes  that  they  are 
to  be  multiplied  together ;  thus, 

7  x  5  =  35 ;    and  is  read,  5  times  1  are  35. 

41.  What  will  4  oranges  cost,  at  2  cents  apiece? — 42.  What  is 
Multiplication  ?— 43.  What  is  the  multiplicand  ?— 44.  What  is  the 
multiplier  ? — 45.  What  is  the  product  ? — 46.  What  are  the  factors  ? 
— 47.  What  is  the  sign  of  Multiplication?  What  is  it  called? 
When  placed  between  two  numbers,  what  does  it  denote  ? 


MULTIPLICATION    OF 


Multiplication  Table. 


IX     1=       1 

2x    1=     2 

3x    1=     3 

4x    1=     4 

1x2=     2 

2X    2=     4 

3x    2=     6 

4x    2=     8 

1x3=     3 

2x    3=     6 

3X    3=     9 

4X    3=   12 

IX    4  =     4 

2X    4=     8 

3X   4=   12 

4x   4=   16 

1X5=     5 

2X    5=   10 

3x    5=   15 

4x    5=   20 

1x6=     6 

2x    6=   12 

3x    6=   18 

4x    6=   24 

lx    7  =     7 

2x    7=   14 

3X    7=  21 

4x    7=  28 

lx    8=     8 

2x    8=   16 

3x    8=   24 

4X    8=  32 

1x9=     9 

2x    9=   18 

3x    9=   27 

4x    9=   36 

1x10=   10 

2x10=   20 

3x10=   30 

4x10=   40 

1  xll=   11 

2x11=   22 

3x11=  33 

4x11=   44 

1x12=   12 

2x12=   24 

3x12=   36 

4x12=   48 

5x    1=     5 

6x    1=     6 

7x    1=     7 

8x    1=     8 

5X    2=   10 

6x    2=   12 

7x    2=   14 

8x    2-   16 

5x    3=   15 

6x    3=   18 

7x    3=   21 

8x    3=   24 

5x    4=  20 

6x    4=   24 

7x   4=   28 

8x    4=  32 

5x    5=  25 

6x    5=   30 

7x    5=  35 

8x    5=  40 

ox    6=   30 

6x    6=   36 

7x    6=  42 

8x    6=  48 

5x    7  =  35 

6x    7=  42 

7x    7=  49 

8x    7=  56 

5x    8=  40 

6x    8=  48 

7x    8=  56 

8x    8=   64 

5x    9=  45 

6x    9=  54 

7x    9=  63 

8x    9=   72 

5x10=  50 

6x10=   60 

7x10=  70 

8x10=   80 

5x11=  55 

6x11=   66 

7x11=  77 

8x11=   88 

5x12=   60 

6x12=   72. 

7x12=   84 

8x12=   96 

9X    1=     9 

10X    1=   10 

11X    1=   11 

12X    1=   12 

9X    2  =   18 

10X    2=   20 

11X    2=   22 

12X   2=   24 

9X   3  =   27 

10X   3=  30 

11X   3=   33 

12X   3=   36 

9X   4=   36 

10X   4=  40 

11X   4=  44 

12X   4=  48 

9X   5=  45 

10X   5=  50 

11X   5=   55 

12X   5=   60 

9X    6=   54 

10X    6=   60 

11X    6=   66 

12X    6=   72 

9x    7=   63 

10X    7=   70 

11X    7=   77 

12X    7=   84 

9x   8=   72 

10X   8=   80 

11X   8=   88 

12X   8=   96 

9x    9=   81 

10X   9=   90 

11  X    9=   99 

12X    9  =  108 

9X10=   90 

10X10  =  100 

11X10  =  110 

12X10=120 

UXll=   99 

10x11  =  110 

11X11  =  121 

12x11  =  132 

9x12=108 

10X12  =  120 

11X12  =  132 

12X12  =  144 

SIM  PUS   NUMBERS.  43 

CASE     I. 

48.   When  the  multiplier  does  not  exceed  9. 
1.  What  is  the  product  of  236  multiplied  by  4? 

OPERATION. 

QQA 

ANALYSIS. — Since  the  entire  number  236  is          J0" 
to  be  taken  4  times,  each  order  of  units  must 
be  taken  4  times:  hence,  the  product  must  24  units, 

contain   24  units,  12   tens,   and   8   hundreds:          12     tens. 

8        hundreds. 

Therefore,  the  product  is 944 

In  practice,  the  operation  is  performed  thus : 
Say,  4  times  6  are  24;  set  down  the  4,  and  then    OPERATION. 
say,  4  times  3  are  12,  and  2  to  carry  are  14;   set 

down  the  4,  and  then  say,  4  times  2  are  8,  and  1          £ 

to  carry  are  9;  set  down  the  9,  and  the  product  is          944 
944,  as  before. 

Hence,  we  have  the  following 

Rule. 

Multiply  each  figure  of  the  multiplicand  by  the  multi- 
plier, carrying  and  setting  down  as  in  Addition. 

49.  Since  the   multiplier  denotes   times,   it  is   always   an 
abstract  number;  and  since  the  repetition  of  a  number  does 
not  change  its  unit,"  the  unit  of  the  product  will  be  the  same 
as  that  of  the  multiplicand. 

Reading. 

50.  The  operations  of  Multiplication  may  be  much  short- 
ened, by  pronouncing  only  the  final  results. 

Thus,  in  the  last  example,  instead  of  saying,  4  times  6  are  24; 
4  times  3  are  12,  and  2  are  14;  4  times  2  are  8,  and  1  are  9: 
we  pronounce  only  the  final  results,  24,  14,  9, — performing  the 
operations  mentally. 


44          „  MULTIPLICATION   OF 

Examples. 

(1.)  (2.)  (3.) 

Multiplicand,    867901  278904  678741 

Multiplier,  123 

(4.)  (5.)  (6.)  (7.) 

47904  780635  910362  1790478 

4567 


8.  Multiply  320  by  1. 

9.  Multiply  9048  by  3. 

10.  Multiply  67049  by  8. 

11.  Multiply  270412  by  5. 


12.  Multiply  30416  by  7. 

13.  Multiply  204127  by  0. 

14.  Multiply  30497  by  6. 

15.  Multiply  69507  by  5. 


16.  If  104  yards  of  cotton  sheeting  are  sufficient  to  sup- 
ply 1   family  for  a  year,  how  many  yards  would  supply  9 
families  ? 

17.  A  farmer  had  309   sheep  in   each  of  4   fields:   how 
many  sheep  had  he  altogether  ? 

18.  Mrs.  Simpkins  purchased  2  rolls  of  table  linen,  each 
containing  149  yards :  how  many  yards  did  she  buy  ? 

19.  Each  of  9  grocers  bought  2974  pineapples :  how  many 
did  they  all  buy? 

20.  If  each  of  7  children  receive  4073  dollars,  how  much 
do  they  all  receive  ?    i^ 

21.  How  many  sheep  are   there   in  9  droves,  if  in  each 
drove  there  are  598  ? 

22.  What  will  be   the   cost   of   9  horses,  at   165  dollars 
apiece  ? 

48.  What  is  the  rule,  when  the   multiplier   contains   but   one 
figure  ? 

49.  What  kind  of  a  number  is  the  multiplier?    What  is  the 
unit  of  the  -product  ? 

50.  How  may  the  operations  of  Multiplication  be  abridged  ?    Give 
an  example. 


111111 
111111 
111111 
111111 
111111 
111111 


SIMPLE    NUMBERS.  45 

51.  The  product  of  two  numbers  is  the  same,  whichever 
be  taken  as  the  multiplier. 

Multiply  any  two  numbers  together;  as,  8  by  6. 

ANALYSIS. — Place  as  many 
1's   in   a   horizontal    row   as  8 

there  are  units  in  the  multi-  , A > 

plicand,  and  make  as  many 
rows  as  there  are  units  in 
the  multiplier :  the  product  is 
equal  to  the  number  of  1's  in 
one  row  taken  as  many  times 
as  there  are  rows ;  that  is,  to 
8  x  6  =  48. 

If  we  consider  the  number  of  Ts  in  a  vertical  row  to  be  the 
multiplicand,  and  the  number  of  rows  the  multiplier,  the  product 
will  be  equal  to  the  number  of  1's  in  a  vertical  row  taken  as 
many  times  as  there  are  vertical  rows ;  that  is,  6x8  =  48. 
Hence, 

The  product  of  two  numbers  is  the  same,  whichever  be 
taken  as  the  multiplier. 


CASE     II. 
52.  When  the  multiplier  contains  two  or  more  figures. 

1.    Multiply  8204  by  603. 

OPERATION. 

ANALYSIS. — The  multiplicand  is  to  be  taken  603  8904 

times.     Taking  it  3  times,  we  obtain  24612.  j^o 

When  we  come  to  take  it  6  hundred  times,  the 

lowest  order  of  units  will  be  hundreds:  hence,  4,  24612 
the  first  figure  of  the  product,  must  be  written  in 

the  third  place.  4947012 

NOTE. — The  product  obtained  by  multiplying  by 
a  single  figure  of  the  multiplier,  is  called,  a  partial  product. 
The  sum  of  the  partial  products,  is  the  required  product. 

51.  Is  the  product  of  two  numbers  altered  by  interchanging  the 
factors  ? 


MULTIPLICATION    OF 


Hence,  we  have  the  following 
Rule. 

I.  Write  the  multiplier  under  the  multiplicand,  placing 
units  of  the  same  order  in  the  same  column. 

II.  Beginning  with  the  units'  figure,  multiply  the-  multi- 
plicand by  each   significant  figure   of  the  multiplier,  and 
write  the  first  figure  of  each  partial  product  directly  under 
its  multiplier. 

III.  Then  add  their  partial  products,  and  the  sum  will 
be  the  required  product. 

Proof. 

Write  the  multiplicand  in  the  place  of  the  multiplier,  and 
find  the  product,  as  before.  If  the  two  products  are  the 
same,  the  work  is  supposed  to  be  right. 


Examples. 


Multiplicand, 
Multiplier, 

(5.) 

46834 
406 

(1.)               (2.)                 (3.) 

365             37864  ^      23293 
84                 209  ^            74 

(6.)                   (7.) 
679084             1098731 
-126                   19S7X, 

9.  Multiply 
10.  Multiply 
11.  Multiply 
12.  Multiply 
13.  Multiply 
14.  Multiply 
15.  Multiply 
16.  Multiply 

12345678  by  32. 
9378964  by  42. 
1345894  by  49. 
576784  by  64. 
596875  by  144. 
46123101  by  72. 
6185720  by  132. 
718328  by  96. 

17.  Multiply  6 
18.  Multiply  8 
19.  Multiply  1 
20.  Multiply  5 
21.  Multiply  5 
22.  Multiply  5 
23.  Multiply  7 
24.  Multiply  1 

(4.) 
47042 
91 

(8.) 

8971432  '- 
10471 

679534  by  9185. 
86972  by  1208. 
1055054  by  279. 
538362  by  9258. 
50406  by  8056. 
523972  by  1527. 
760184  by  1615. 
105070  by  3145. 


NTMHKliS.  47 


25.  What  Is  the  product  of  the  number  728056,  multiplied 
by  50467  ?  ^   , 

26.  What  is  the   product   of  the   number  579073,   taken 
604678  times? 

27.  What  will  be  the  result  of  taking  the  number  590587, 
79904  times? 

28.  If  the  number  9127089  be  taken  670456  times,  what 
number  will  express  the  result  ? 

29.  What  is  the  product,  when  the  number  30726  is  mul- 
tiplied by  97034219? 

30.  What  is  the  product  of  the  two  numbers,  870623  and 
91678538? 

31.  Multiply  five  thousand  nine  hundred  and  sixty-five,  by 
six  thousand  and  nine. 

32.  Multiply  eight  hundred  and  seventy  thousand  six  hun- 
dred and  fifty-one,  by  three  hundred  and  seven  thousand  and 
four. 

33.  Multiply  four  hundred  and  sixty-two  thousand  six  hun- 
dred and  nine,  by  itself. 

34.  Multiply  eight  hundred  and  forty-nine  millions  six  hun- 
dred and  seven  thousand  three  hundred  and  six,  by  nine  hun- 
dred thousand  two  hundred  and  four. 

35.  Multiply  704  millions  130  thousand  496,  by  three  thou- 
sand three  hundred  and  one. 

36.  Multiply  forty-nine  millions  forty  thousand  six  hundred 
and  ninety-seven,  by  nine  millions  forty  thousand  seven  hun- 
dred and  nine. 

Composite  Numbers.  —  Factors. 

53.  A  COMPOSITE  NUMBER  is  one  which  may  be  produced 
by  multiplying  together  two  or  more  numbers. 

54.  A  FACTOR  is  any  one  of  the  numbers  which,  multiplied 
together,  produce  a  composite  number. 

Thus,  2x3  =  6,  2  and  3  are  the  factors  of  the  composite 
number  6. 

53.  What  is  a  composite  number?  —  54.  What  is  a  factor? 


48  MULTIPLICATION   OF 

\ 

Also,  12  =  6x-2  =  3x2x2,  is  a  composite  number, 
and  the  factors  are  6  and  2  ;  but  6  is  a  composite  number, 
whose  factors  are  3  and  2  :  hence,  3,  2,  and  2  are  factors 
of  12. 

1.  What  are  the  factors  of  8  ?     Of  9  ?     Of  10  ?     Of  14  ? 

2.  What  are  the  factors  of  4  ?     Of  28  ?     Of  30  ?     Of  32  ? 

55.   When    the    multiplier    is  a   composite   number. 

1.  Multiply  8  by  the  composite  number  6,  of  which  the 
factors  are  3  and  2. 

8 


1 
1 

1 
1 

1 
1 

1 

1 

1 

1 

1 
1 

1 
1 

}}8X 

2 

=  16 

1 
1 

1 
1 

1 
1 

1 
1 

1 
1 

1 
1 

1 
1 

1 
1 

8  x 

2 

=  16 

1 
1 

1 
1 

1 
1 

1 
1 

1 

1 

1 
1 

1 
1 

1 
1 

•  8X 

2 

=  16 

48 

24 

_2 

48 

ANALYSIS. — If  we  write  6  horizontal  lines,  with  8  units  in  each, 
the  product  of  8x6=  48,  will  express  the  number  of  units  in 
all  the  lines. 

If  we  divide  the  horizontal  lines  into  sets  of  3  each  (as  on  the 
left),  there  will  be  2  sets;  the  number  in  each  set  will  be 
8x3  =  24,  and  since  there  are  2  sets,  the  whole  number  of  units 
wiifbe  24x2  =  48. 

If  we  divide  the  lines  into  sets  of  2  each  (as  on  the  right), 
there  will 'be  3  sets;  the  number  in  each  set  will  be  8x2  =  16, 
and  since  there  are  3  sets,  the  whole  number  of  units  will  be 
16  x  3  =  48.  Hence, 

If  the  multiplier  is  a  composite  number,  multiply  by  the 
factors,  in  succession. 

Contractions  in  Multiplication. 

56.  CONTRACTIONS,  in  Multiplication,  are  short  methods  of 
finding  the  product,  when  the  multiplier  is  a  composite  number. 


55.  How  do  you  multiply,  when  the  multiplier  is  a  composite 
number  ? 

50.  What  are  contractions,  in  Multiplication? 


6.  Multiply  91738  by  81. 

7.  Multiply  3842  by  144. 

II. 


SIMPLK    NUMBERS.  49 

CASE     I. 
57.    When  the  factors  are  any  numbers. 

Rule. 

I.  Separate  the  composite  number  into  its  factors. 

II.  Multiply  the   multiplicand   by   one  factor,   and   thf 
product  by  a  second  factor ;  and  so  on,  till  all  the  factor^ 
have  been  used:  the  last  product  will  be  the  product  required. 

Examples. 

*>     1.    Multiply  327  by  12. 

The  factors  of  12  are  2  and  6 ;  they  are  also  3  and  4,  or 
they  are  3,  2,  and  2. 

For,     2  x  6  =  12,     3  x  4  =  12,     and     3  x  2  x  2  =  12. 

X    2.  Multiply  5709  by  48.         5.  Multiply  937387  by  54. 
V  3.  Multiply  342516  by  5G. 
4.  Multiply  209402  by  72. 

CASE    II. 

58.  When  the  multiplier  is  1,  with  any  number  of  ciphers 
annexed;  as,  10,  100,  1000,  &c. 

Placing  a  cipher  on  the  right  of  a  number,  is  called,  an- 
nexing it.  Annexing  one  cipher,  increases  the  unit  of  each 
place  ten  times :  that  is,  it  changes  units  into  tens,  tens  into 
hundreds,  hundreds  into  thousands,  &c. ;  and,  therefore,  in- 
creases the  number  ten  times. 

Thus,  the  number  5  is  increased  ten  times  by  annexing 
one  cipher,  which  makes  it  50.  The  annexing  of  two  ciphers 
increases  a  number  one  hundred  times ;  the  annexing  of 
three  ciphers,  a  thousand  times,  &c. :  hence,  the  following 

Rule. 

Annex  to  the  multiplicand  as  many  ciphers  as  thei*e  are 
in  the  multiplier,  and  the  number  so  formed  will  be  the 
required  product. 

57.  How  do  you  multiply,  when  the  factora**re  any  numbers  ? 

V  c 


50 


MULTIPLICATION    OF 


1.  Multiply  254  by  10. 

<  2.  Multiply  648  by  100. 

-N3.  Multiply  7987  by  1000. 

-4.  Multiply  9840  by  10000. 


Examples. 

5.  Multiply  3750  by  100. 

6.  Multiply  6704  by  10000. 

7.  Multiply  2141  by  100. 

8.  Multiply  872  by  100000. 


CASE    III. 

59.  When  there  are  ciphers  on  the  right  hand  of  one  or 
both  of  the  factors. 

In  this  case,  each  number  may  be  regarded  as  a  composite 
number,  of  which  the  significant  figures  are  one  factor,  and  1, 
with  the  requisite  number  of  ciphers  annexed,  the  other. 

1.    Let  it  be  required  to  multiply  3200  by  800. 

OPERATION. 

3200  =  32  x  100 ;     and     800  =  8  x  100. 
Then,     3200  x  800  =  32  x  100  x  8  x  100, 
=  32  x  8  x  100  x  100, 
=  2560000. 
Hence,  we  have  the  following 

Rule. 

Omit  the  ciphers,  and  multiply  the  significant  figures : 
then  place  as  many  ciphers  at  the  right  hand  of  the  product 
as  there  are  in  both  factors. 

Examples. 


(1.) 

(2.) 

(3.) 

76400 

7532000         416000 

24 

580 

357000 

1833600 

148512000000 

4. 

4871000 

X  270000.  > 

7.  21200  x  70. 

5. 

296200  x  875000. 

8.  359260  x  304000. 

6. 

3456789 

X  567090. 

9.  7496430  x  695000. 

58.  If  you  place  one  cipher  on  the  right  of  a  number,  what  effect 
has  it  on  its  value  ?  If  you  place  two,  what  effect  lias  it  ?  If  you 
place  three  ?  How  mucli  will  each  increase  it  ?  How  do  you  mul- 
tiply by  10,  100,  1000,  &c.? 

69.  When  there  are  ciphers  on  the  right  hand  of  one  or  both  the 
factors,  how  do  you  multiply  ? 


PRACTICAL    EXAMPLES. 

Applications  in  the  preceding  Huies. 

60.   1.  What  will  5  yards  of  cloth  cost,  at  7  dollars  a  yard? 

ANALYSIS. — Five  yards  of  cloth  will  cost  5  times  as  much  as  1 
yard ;  since  1  yard  of  cloth  costs  7  dollars,  5  yards  will  cost  5 
times  7  dollars,  which  are  35  dollars :  Therefore,  5  yards  of  cloth, 
at  7  dollars  a  yard,  will  cost  35  dollars:  hence, 

The  cost  of  any  number  of  things,  is  equal  to  the  price 
of  a  single  thing  multiplied  by  the  number. 

We  have  seen  that  the  product  of  two  numbers  will  be  the 
same  (that  is,  will  contain  the  same  number  of  units),  which- 
ever be  taken  for  the  multiplicand  (Art.  51).  Hence,  in 
practice,  we  may  multiply  the  two  factors  together,  taking 
either  for  the  multiplier,  and  then  assign  the  proper  unit  to 
the  product.  We  generally  take  the  least  number  for  the 
multiplier. 

2.  There  are  ten  bags  of  coffee,  each  containing  48  pounds  : 
how  much  coffee  is  there  in  all  the  bags  ? 

3.  There  are  24  hours  in  a  day,  and  7  days  in  a  week : 
how  many  hours  in  a  week  ? 

4.  A   merchant   bought   49   hogsheads   of  molasses,   each 
containing  63  gallons  :   how  many  gallons  of  molasses  were 
there  in  the  parcel  ? 

5.  If  a  regiment  of  soldiers  contains  1128  men,  how  many 
men  are  there  in  an  army  of  106  regiments  ? 

6.  A  merchant  -buys  a  piece  of  cloth  containing  97  yards, 
at  3  dollars  a  yard:  what  does  the  piece  cost  him? 

7.  Suppose  a  man  were  to  travel  32  miles  a  day:   how 
far  would  lie  travel  in  365  clays  ? 

8.  There  are  20  pieces  of  cloth,  each  containing  37  yards, 
and  49  other  pieces,  each  containing  75  yards  :    how  many 
yards  of  cloth  are  there  in  all  the  pieces  ? 

9.  A  farmer  bought  a  farm  containing  10  fields;  three  of 
he  fields  contained  9  acres  each ;  three  other  of  the  fields, 
2  acres  each  ;   and  the  remaining  4  fields,  each  15  acres : 

60.  How  do  you  find  the  cost  of  any  number  of  things?  How 
may  it  be  done  in  practice? 


52  PRACTICAL   EXAMPLES. 

how  man 


the  whole  cost,  at  18  dollars  an  acre  ? 

10.  In  a  certain   city,   there   are   3*151  houses.     If  each 
house,  on  an  average,  contains  5  persons,  how  many  inhabi- 
tants are  there  in  the  city? 

11.  If  186  yards  of  cloth  can  be  made  in  one  day,  how 
many  yards  can  be  made  in  1252  days  ? 

12.  If  30009   cents   are  paid  for  one   man's   labor  on  a 
railroad  for  1  year,  how  many  cents  would  be  paid  to  814 
men,  each  man  receiving  the  same  wages  ? 

13.  There  are   320   rods  in  a  mile  :   how  many  rods  are 
there  in  the  distance  from  St.  Louis  to  New  Orleans,  which 
is  1092  miles  ? 

14.  Suppose   a  book   to   contain  470  pages,   45  lines  on 
each  page,  and  50  letters  in  each  line  :  how  many  letters  in 
the  book? 

15.  Supposing  a  crew  of  250  men  to  have  provisions  for 
30  days,  allowing  each  man   20  ounces  a  day  :    how  many 
ounces  have  they? 

16.  There  are  350  rows  of  trees  in  a  large  orchard,  125 
trees  in  each  row,  and  3000  apples  on  each  tree  :  how  many 
apples  in  the  orchard  ? 

17.  How  many  soldiers  are  there  in  a  body  of  men  that 
is  7  deep,  if  each  line  has  758  men  ? 

18.  If  a  railroad  car  goes  27  miles  an  hour,  how  far  will 
it   run   in  3   days,   running  20   hours   each  day?     How  far 
would  it  run,  if  its  rate  were  37  miles  an  hour? 

19.  If  1327  barrels  of  flour  will  feed  the  inhabitants  of  a 
city  for  1  day,  how  many  barrels  will  supply  them  for  2  years  ? 

20.  A  regiment  of  men  contains  10  companies,  each  com- 
pany 8  platoons,  and  each  platoon  34  men  :  how  many  men 
in  the  regiment  ? 

21.  Two  persons  start  from  the  same  place,  and  travel  in 
the  same  direction  ;   one  travels  at  the  rate  of  6  miles  an 
hour,  the  other  at  the  rate  of  9  miles  an  hour  :  if  they  travel 
8  hours  a  day,  how  far  will  they  be  apart  at  the  end  of  17 
days  ?     How  far,  if  they  travel  in  opposite  directions  ? 


PRACTICAL   EXAMPLES.  53 

22.  The  Erie  railroad  is  about  425  miles  long,  and  cost 
65   thousand   dollars  a  mile  :    if  9645635  dollars  had  been 
paid,  how  much  would  remain  unpaid  ? 

23.  A  drover  bought  106  oxen,  at  35  dollars  a  head ;  it 
cost  him  6  dollars  a  head  to  get  them  to  market,  where  he 
sold  them  at  41  dollars  :  did  he  make  or  lose,  and  how  much? 

24.  The  great  Illinois  Central  railroad  reaches  from  Chi- 
cago to  the  mouth  of  the  Ohio  river,  815  miles,  and  cosfc 
23500  dollars  a  mile :  what  was  its  entire  cost  ? 

Bills  of  Parcels. 

61.  When  a  person  sells  goods,  he  generally  gives  with  them 
a  bill,  showing  the  amount  charged  for  them,  and  acknowl- 
edging the  receipt  of  the  money  paid ;  such  bills  are  called, 

Hills  of  Parcels. 

*>  NEW  YORK,  Oct.  1,  1854. 

25.  James  Johnson  Bought  of  W.  Smith. 
4  Chests  of  tea,  of  45  pounds  each,  at  1  doll,  a  pound 

3  Firkins  of  butter,  at  17  dolls,  per  firkin 

4  Boxes  of  raisins,  at  3  dolls,  per  box 

3(5  Bags  of  coffee,  at  16  dolls,  each        .... 
14  Hogsheads  of  molasses,  at  28  dolls,  each 

Amount,  dollars. 

Received  the  amount  in  full.         W.  SMITH. 


HARTFORD,  Nov.  1,  1854. 

26.     James  Hughes  Bought  of  W.  Jones. 

27  Bags  of  coffee,  at  14  dollars  per  bag 
18  Chests  of  tea,  at  25  dolls,  per  chest 
75  Barrels  of  shad,  at  9  dolls,  per  barrel     . 
87  Barrels  of  mackerel,  at  8  dolls,  per  barrel    . 

67  Cheeses,  at  2  dolls,  each 

59  Hogsheads  of  molasses,  at  29  dolls,  per  hogshead 

Amount,  dollars. 

Received  the  amount  in  full,  for  W.  JONES, 

per  James  Cross. 

61.  What  are  bills  of  parcels  ? 


64-  DIVISION   OF 


DIVISION. 

62.  1.  When  a  number  is  divided  into  2  equal  parts,  each 
part  is  called,  one-half  of  the  number. 

What  is  one-half  of  4  apples  ?     What  is  one-half  of  4  ? 
How  many  times  is  2  contained  in  4  ? 

2.  When  a  number  is  divided  into  3  equal  parts,  each  part 
is  called,  one-third  of  the  number. 

What  is  one-third  of  9  apples  ?     What  is  one-third  of  9  ? 
How  many  tunes  is  3  contained  in  9  ? 

3.  When  a  number  is  divided  into  4  equal  parts,  each  part 
is  called,  one-fourth  of  the  number. 

What  is  one-fourth  of  12  pears  ?    What  is  one-fourth  of  12  ? 
How  many  times  is  4  contained  in  12?  Ar 

4.  When  a  number  is  divided  into  5  equal  parts,  each  part 
is  called,  one-fifth  of  the  number. 

What  is  one-fifth  of  ten  marbles  ?     What  is  one-fifth  of  10  ? 
How  many  tunes  is  5  contained  in  10  ? 

5.  When  a  number  is  divided  into  6  equal  parts,  each  part 
is  called,  one-sixth  of  the  number. 

6.  If  12  apples   be   equally  divided   among  4  boys,  how 
many  will  each  have  ? 

ANALYSIS. — If  12  apples  be  divided  equally  among  4  boys,  'each 
boy  will  have  one  of  the  four  equal  parts  of  12  apples :  one  of  the 
4  equal  parts  of  12  is  3  :  Therefore,  each  boy  will  have  3  apples. 

T.  If  24  peaches  are  divided  equally  among  6  boys,  how 
many  will  each  have?  What  is  one  of  the  six  equal  parts 
of  24  peaches  ?  How  many  times  is  6  contained  in  24  ? 

8.  How  many  yards  of  cloth,  at  3  dollars  a  yard,  can  you 
buy  for  24  dollars  ? 

62.  What  is  one-half  of  a  number?  What  is  one-third  of  a  num- 
ber ?  What  is  one-fourth  of  a  number  ?  What  is  one-fifth  of  a 
number  ? 


SIMPLE   NUMBERS.  55 

ANALYSIS. — Since  1  yard  costs  3  dollars,  you  can  buy  as  many 
yards  as  3  is  contained  times  in  24 :  3  is  contained  in  24,  8  times : 
Therefore,  at  3  dollars  a  yard,  you  can  buy  8  yards  of  cloth  for 
24  dollars. 

9.  A  farmer  pays  28  dollars  for  7  sheep:   how  much  is 
that  apiece  ? 

ANALYSIS. — Since  7  sheep  cost  28  dollars,  one  sheep  will  c'os 
as  many  dollars  as  7  is  contained  times  in  28,  which  is  4 
Therefore,  each  sheep  will  cost  4  dollars. 

10.  If  12  yards  of  muslin  cost  96  cents,  how  much  does  1 
yard  cost  ? 

11.  How  many  oranges  could  you  buy  for  72  cents,  if  they 
cost  6  cents  apiece? 

63.  DIVISION  is  the  operation  of  dividing  a  number  into 
equal  parts  ;   or,  of  finding  how  many  times  one  number  is 
contained'  in  another. 

64.  The  DIVIDEND  is  the  number  to  be  divided. 

65.  The  DIVISOR  is  the  number  by  which  we  divide  :    it 
shows  into  how  many  equal  parts  the  dividend  is  divided. 

66.  The  QUOTIENT  is  the  result  of  division.     It  is  one  of  the 
equal  parts  of  the  dividend,  and  if  the  numbers  have  the  same 
unit,  shows  how  many  times  the  dividend  contains  the  divisor. 

67.  The  REMAINDER  is  what  is  left  after  the  operation. 

68.  EXACT  DIVISION  is  when  the  remainder  is  0. 

69.  There  are  three  signs  used  to  denote  Division  : 
18  -h  4,     expresses  that  18  is  to  be  divided  by  4. 

•I   Q 

_ ,  expresses  that  18  is  to  be  divided  by  4. 

4 

4)  18,        expresses  that  18  is  to  be  divided  by  4. 

63.  What  is  Division  ?— 64.  What  is  the  dividend? — 65.  What  is 
the  divisor? — 66.  What  is  the  quotient?  What  does  the  quotient 
show  ? — 67.  What  is  the  remainder  ?— 68.  What  is  exact  division  ? 
— 69.  How  many  signs  are  there  of  Division  ?  Write  them. 


56 


DIVISION    OF 


CASE     I. 
.70.   "When  the  divisor  is  less  than  10. 

1.    Divide  86  by  2.  OPERATION. 

ANALYSIS.  —  There  are  8  tens  and  6  units          ^     ^- 
to^be  divided  by  2.     We  say,  2  in  8,  4  times,          |     r| 
which  being  tens,  we  write   it  in  the  tens 
place.     We  then  say,  2  in  6,  3  times,  which 
being  units,  are  written  in  the  units  place. 
Hence,  the  quotient  is  43. 


>  > 
P  S 
2)86 

43  quotient. 


2.  Divide  466  by  8. 

ANALYSIS.  —  We  first  divide  the  46  tens  by 
8,  giving  a  quotient  of  5  tens,  and  6  tens 
over.  These  6  tens  are  equal  to  60  units,  to 
which  add  the  6  in  the  units  place.  Then 
say,  8  in  66,  8  times  and  2  over:  hence, 
the  quotient  is  58,  and  a  remainder  of  2. 
This  remainder  is  written  after  the  last  quo- 
tient figure,  and  the  8  placed  under  it;  the 
quotient  is  read,  58  and  2  divided  by  8. 

3.  Let  it  be  required  to  divide  30456  by  8. 

ANALYSIS.  —  We  first  say,  8  in  3  we  can  not. 
Then,  8  in  30,  3  times  and  6  over  ;  then,  8  in  64, 
8  times;  then,  8  in  5,  0  times;  then,  8  in  56, 
7  times. 

Hence,  we  have  the  following 


OPERATION. 

8)466 

58-2  rein. 

58|  quot. 


OPERATION. 

8 )  30456 
3807 


Rule. 

I.  Write  the  divisor  on  the  left  of  the  dividend.     Begin- 
ning at  the  left,  divide  each  figure  of  the  dividend  by  the 
divisor,  and  set  each  quotient  figure  under  its  dividend. 

II.  If  there  is  a  remainder  after  any  division,   annex 
to  it  the  next  figure  of  the  dividend,  and  divide  as  before. 

III.  If  any  dividend  is  less  than  the  divisor,  write  0  for 
the  quotient  figure,  and  annex  the  next  figure  of  the  divi- 
dend, for  a  new  dividend. 


SIMPLE   NUMBERS.  57 

IV.  If.  there  is  a  remainder,  after  dividing  the  last  fig- 
ure, set  the  divisor  under  it,  and  annex  the  result  to  the 
quotient. 

Proof. 

Multiply  the  entire  part  of  the  quotient  by  the  divisor, 
and  to  the  product  add  the  remainder  :  if  the  work  is  right, 
the  result  will  be  equal  to  the  dividend. 


(3.) 
4  )  73684 

Ans. 

Proof,       9369  825467 

(4.)  (5.)  (6.) 

5)673420  7)446396  5)1746809 


(1.) 

3)9369 

Examples. 

(2.) 
6)825467 

3123 
3 

J37577| 
6 

A  7.  Divide  $6434  by  2. 
V  8.  Divide  416710  by  4. 

9.  Divide  64140  by  5. 

10.  Divide  278943  by  6. 

fll.  Divide  95040522  by  6. 

X12.  Divide  75890496  by  8. 

13.  Divide  6794108  by  3. 

14.  Divide  21090431  by  9. 


• 


15.  Divide  2345678964  by  6. 

16.  Divide  570196382  by  7. 

17.  Divide  67897634  by  9. 

18.  Divide  75436298  by  8. 

19.  Divide  674189904  by  9. 

20.  Divide  1404967214  by  5. 

21.  Divide  27478041  by  7. 

22.  Divide  167484329  by  3. 


23.  If  it  takes  5  bushels  of  wheat  to  make  a  barrel  ot 
flour,  how  many  barrels  can  be  made  from  65890  bushels? 

24.  How  many  barrels  of  flour,  at  7  dollars  a  barrel,  can 
be  bought  for  609463  dollars? 

25.  A  vessel  sails  8  miles  an  hour :   in  how  many  hours 
will  it  sail  3756  miles  ? 


70.   Give  the  rule  for  Division,  when  the  divisor  is  less  than  10. 
How  do  you  prove  Division  ? 

8* 


58  DIVISION   OF 

26.  Suppose  a  wheel  is  7  feet  in  circumference  :  how  many 
times  would  it  turn,  in  going  15840  feet  ? 

27.  If  a  pace  is  3  feet,  how  many  paces  will  a  man  take 
in  walking  6  miles,  or  31680  feet  ? 

28.  When  John  starts,  Joseph  is  37594  feet  ahead;  Joseph 
goes  251  feet  a  minute,  and  John  goes  260  feet  a  minute: 
in  how  many  minutes  will  John  overtake  Joseph  ? 

29.  A  county  contains  207360  acres  of  land,  lying  in  9 
townships  of  equal  extent :  how  many  acres  in  each  township  ? 

30.  An  estate,  worth  2943  dollars,  is  to  be  divided  equally 
among  a  father,  mother,  3  daughters,  and  4  sons :  what  is 
the  portion  of  each  ? 

31.  A  railroad,  worth  544806  dollars,  is  owned,  in  equal 
shares,  by  9  persons :  what  is  the  value  of  the  share  of  each  ? 

CASE    II. 
71.     "When  the  divisor  exceeds  9.  Q 

1.    Let  it  be  required  to  divide  7059  by  13. 

ANALYSIS. — The  divisor,  13,  is  not 

contained  in  7  thousands;   therefore,  OPERATION. 

there  are  no  thousands  in  the  qiiotient.  m  ^    ,  ^     g    .  & 

We  then  consider  the  0  to  be  an-  J  §  f  B      §  1  '3 

nexed  to  the  7,  making  70  hundreds,  &  W  H  t>     W  H  t> 

and  call  this  a  partial  dividend.  13)7059(543 

The  divisor,  13,  is  contained  in  70  6  5 

hundreds,  5  hundreds  times  and  some-  5  5 

thing  over.     To  find  how  much  over,  5  2 

multiply  13  by  5  hundreds,  and  sub- 
tract the  product,  65,  from  70,  and 
there  will  remain  5  hundreds,  to  which 
bring  down  the  5  tens,  and  consider  0 

the  55  tens  a  new  partial  dividend. 

Then,  13  is  contained  in  55  tens,  4  tens  times  and  something 
over.  Multiply  13  by  4  tens,  and  subtract  the  product,  52,  from 
55,  and  to  the  remainder,  3  tens,  bring  down  the  9  units,  and 
consider  the  39  units  a  new  partial  dividend. 

Then,  13  is  contained  in  39,  3  times.  Multiply  13  by  3,  and 
subtract  the  product,  39,  from  89,  and  we  find  that  the  remain- 
der is  0. 


SIMPLE   NUMBERS.  59 

2.   Let  it  be  required  to  divide  2756  by  26. 

"We  first  say,  26  in  27,  once,  and  place  1 
in  the  quotient.     Multiplying  by  1,  subtract-  OPERATION. 

ing,  and  bringing  down  the  5,  we  have  15         26)2756  (106 
for  the  first  partial  dividend.     We  then  say,  26 

26  in  15,  0  times,  and  place  the  0  in  the  TKA 

quotient.     We  then  bring  down  the  6,  and  ,lfi 

find   that  the  divisor  is    contained  in  156, 
6  times. 

Hence,  if  any  one  of  the  partial  dividends  is  less  than  the 
divisor,  write  0  for  the  quotient  figure,  and  bring  down  the  next 
figure,  forming  a  new  partial  dividend. 

Rule. 

I.  Write  the  divisor  on  the  left  of  the  dividend. 

II.  Note  the  fewest  figures  of  the  dividend,  at  the  left, 
that  will  contain  the  divisor,  and  set  the  quotient  figure  at 
the  right  of  the  dividend. 

III.  Multiply  the  divisor  ly  the  quotient  figure,  subtract 
the  product  from  the  first  partial  dividend,  and  to  the  re- 
mainder annex  the  next  figure  of  the  dividend,  forming  a 
second  partial  dividend. 

IY.  Find,  in  the  same  manner,  the  second  and  succeed- 
ing figures  of  the  quotient,  till  all  the  figures  of  the  dividend 
are  brought  down. 


1.  —  There  are  five  operations  in  Division:  1st.  To  wrijbe 
down  the  numbers;  2d.  Divide,  or  find  how  many  times;  3d. 
Multiply;  4th.  Subtract;  5th.  Bring  down,  to  form  the  partial 
dividend. 

2.  The  product  of  a  quotient  figure  by  the  divisor  must  never 
be  larger  than  the  corresponding  partial  dividend;   if  it  is,  the 
quotient  figure  is  too  large,  and  must  be  diminished. 

3.  When  any  one  of  the  remainders  is  greater  than  the  divisor, 
the  quotient  figure  is  too  small,  and  must  be  increased. 

4.  The  unit  of  any  quotient  figure  is  the  same  as  that  of  the 
partial  dividend  from  which  it  is  obtained.    The  pupil  should 
always  name  the  unit  of  e.very  quotient  figure. 

5.  The  unit  of  a  remainder  is  the  same  as  that  of  the  dividend. 


60  DIVISION   OF 

Proof. 

72.  In  Division,  the  divisor  shows  into  how  many  equal 
parts  the  dividend  is  divided  :  the  quotient  is  one  of  these 
parts,  and  the  remainder  is  what  is  left. 

Hence,  to  prove  Division, 

Multiply  the  divisor  by  the  quotient,  and  to  the  product 
add  the  remainder.  If  the  work  is  right,  the  sum  will  be 
the  same  as  the  dividend. 

Examples. 

1.  If  300  be  divided  into  60  equal  parts,  what  is  one  of 
these  parts  ? 

2.  How  many  times  is  54  contained  in  7574  ? 

3.  If  295470  be  divided  into  90  equal  parts,  what  is  one 
of  these  parts  ? 

^&.   How  many  times  is  37  contained  in  7210449? 

5.  If  62205  dollars  be  divided  equally  among  a  regiment 
consisting  of  957  men,  how  many  dollars  will  each  have  ? 

6.  What  is  one  of  the  equal  parts  of  the  number  66708, 
when  divided  by  204  ? 


71.  What  is  the  rule  for  division,  when  the  divisor  exceeds  9? 

NOTES. — 1.   How  many  operations  are  there  in  Division  ?    Name 
them. 

2.  If  a  partial  product  i?  greater  than  the  partial  dividend,  what 
does  it  indicate. 

3.  What  do  you  do  when  any  one  of  the  remainders  is  greater 
than  the  divisor? 

4.  What  is  the  unit  of  any  figure  of  the  quotient  ?    When  the 
divisor  is  contained  in  simple  units,  what  will  be  the  unit  of  the 
quotient  figure?    When  it  is  contained  in  tens,  what  will  be  the 
unit  of  the  quotient  figure?    When  it  is  contained  in  hundreds? 
In  thousands  ? 

5.  What  is  the  unit  of  the  remainder? 

72.  In  Division,  what  does  the  divisor  show  ?  What  the  quotient  ? 
What  is  the  remainder  ?    How  do  you  prove  Division  ? 


SIMPLE  NUMBERS.  61 

7.  How  many  times   is  the  number  43   contained  in  the 
number  12986? 

8.  How  many  times   is  the  number  627  contained  in  the 
number  657723? 

9.  What  is  one  of  the  equal  parts  of  256  barrels  of  flour, 
divided  equally  among  16  families  ? 

10.  How  many  times  is  the  number  804  contained  in  the 
number  320796? 


16.  Divide  14420946  by  74. 

17.  Divide  295470  by  90. 
,18.  Divide  1874774  by  162. 

Divide  435780  by  216. 


>(!!.  Divide  147735  by  45. 

12.  Divide  947387  by  54. 

13.  Divide  145260  by  108. 

^  14.  Divide  79165238  by  238. 

'    15.  Divide  62015735  by  78.    460.  Divi.  119836687  by  3041. 
3%  21.    Divide  203812983  b/5049. 
T  22.   Divide  20195411808  by  3012. 

23.  Divide  74855092410  by  949998. 

24.  Divide  47254149  by  4674. 

25.  Divide  119184669  by  38473. 

26.  Divide  280208122081  by  912314. 

27.  Divide  293839455936  by  8405.>/ 

28.  Divide  4637064283  by  57606.-*^ 

29.  Divide  352107193214  by  210472.  u 

30.  Divide  558001172606176724  by  2708630425.  •'•' 

31.  Divide  1714347149347  by  57143. ^ 

32.  Divide  6754371495671594  by  678957.  *-*j 

33.  Divide  71900715708  by  37149. 

34.  Divide  571943007145  by  37149. 

35.  Divide  671493471549375  by  47143.  "7 

36.  Divide  571943007645  by  37149. 

37.  Divide  171493715947143  by  57007. 

38.  Divide  121932631112635269  by  987654321. 

39.  In  a  hogshead  there  are  63  gallons :  how  many  hogs- 
heads are  there  in  a  reservoir,  containing  2645750  gallons  ? 

40.  A  drover  wishes  to  divide  15600  cattle  into  75  droves 
how  many  cattle  must  he  put  in  each  drove  ? 


62  DIVISION   OF 

73.   Principles  resulting  from  Division. 

1.  When  the  divisor  is  1,  the  quotient  will  be  equal  to 
the  dividend. 

2.  When  the  divisor  is  equal  to  the  dividend,  the  quotient 
will  be  1. 

3.  When  the  divisor  is  less  than  the  dividend,  the  quotien 
will  be  greater  than  1. 

4.  When  the  divisor  is  greater  than  the  dividend,  the  quo- 
tient will  be  less  than  1. 

Proof  of  Multiplication. 

74.  In  Division,  the  divisor  and  quotient  are  factors  of 
the  dividend.  In  Multiplication,  the  multiplicand  and  multi- 
plier are  factors  of  the  product :  Hence, 

If  the  product  of  two  numbers  be  divided  by  the  multipli- 
cand, the  quotient  will  be  the  multiplier ;  or,  if  the  product 
be  divided  by  the  multiplier,  the  quotient  witt  be  the  mul~ 

Examples. 

3679  Multiplicand.  3679)1203033(321 

327  Multiplier.  11037 


25753  9933 

7358  7358 


25753 
1203033  Product.  25753 


73.  When  the  divisor  is  1,  what  is  the  quotient?     When  the 
divisor  is  equal  to  the  dividend,  what  is  the  quotient?    When  the 
divisor  is  less  than  the  dividend,  how  does  the  quotient  compare 
with  1  ?    When  the  divisor  is  greater  than  the  dividend,  how  does 
the  quotient  compare  with  1  ? 

74.  In  Multiplication,  what  are  the  factors  of  the  product  ?    If  the 
product  be  divided  by  the  multiplicand,  what  is  the  quotient  ?    If  it 
be  divided  by  the  multiplier,  what  is  the  qu'otftfnt  ? 


SIMPLE   NUMBERS.  63 

2.  The  product  of  two  factors  is  68959488  ;   one  factor, 
96  ;   what  is  the  other  ? 

3.  The    multiplier   is    270000  ;   now,    if    the    product    be 
1315170000000,  what  will  be  the  multiplicand? 

Contractions  in  Division. 

75.  CONTRACTIONS  IN  DIVISION,  are  short  methods  of  find- 
ing the  quotient,  when  the  divisor  is  a  composite  number. 

CASE    I. 
76.     "When  the  divisor  is  any  composite  number. 

1.  Let  it  be  required  to  divide  1407  dollars  equally  among 
21  men.  Here  the  factors  of  the  divisor  are  7  and  3. 

ANALYSIS. — Let  the   1407  dollars 

be  first  divided  into  7  equal  piles.  OPERATION. 

Each  pile  will  contain  201  dollars.       7  \  1407 
Let  each  pile  be  now  divided  into 

3  equal  parts.     Each  part  will  con-          3  )  2Q1  lst  quotient, 
tain  67  dollars,  and  the  number  of  (J7  quotient  sought, 

parts  will  be   21 :    hence   the   fol- 
lowing 

Rule. 

Divide  the  dividend  by  one  of  the  factors  of  the  divisor ; 
then  divide  the  quotient,  thus  arising,  by  a  second  factor, 
and  so  on,  till  every  factor  has  been  used  as  a  divisor;  the 
last  quotient  will  be  the  answer. 

Examples. 

Divide  the  following  numbers  by  the  factors  : 


1.  1260  by  12  =  3x4. 

2.  18576  by  48  =  4x12.  " 

3.  9576  by  72  =  9x8. 

4.  19296  by  96  =  12x8. 


5.  55728  by  4x9x4  =  144. 

6.  92880  by  2x2x3x2x2. 

7.  57888  by  4x2x2x2. 

8.  154368  by  3x2x2. 


75.  What  are  contractions,  in  Division?    What  is  a  composite 
number  ? 

76.  What  is  the  rule,  when  the  divisor  is  any  composite  number  ? 


64:  DIVISION   OF 


True  Remainder,  when  the  divisor  is  a  composite  number. 

Let  it  be  required  to  divide  755  grapes  into  24  equal 
parts.  24  =  2  x  3  x  4. 

ANALYSIS.  —  If  T55  grapes  be          0^*755 
divided  into  2  equal  parts,  there  )  - 

will  be  377  bunches,  each  con-         3)377     .     .     1,     1st  rein. 
taining   2   grapes,   and   1  grape  "~ 

over.      The   unit   of   377,   is   1          4  >  JLfl    '     '     *> 
bunch  =  2  grapes.    The  unit  of  31     .     .     1,     3d  rem. 

the  first  remainder,  is  the  same  ^  renh  _  ^ 

as  that  of  the  dividend  :  hence,  2x2  =4 

that  remainder  forms  a  part  of 
the  true  remainder.  5  x/     '     '      ~_2 

If  we  divide  377  bunches  into  True  remainder,     11 

3   equal    parts,    we    shall   have 

125    piles,    each    containing    3    bunches,    and    2    bunches    over 
=  2x2  =  4  grapes. 

If  we  divide  125  piles  into  4  equal  parts,  we  shall  have  31 
new  piles,  and  1  pile  over  =3x2  =  6  grapes.  Hence,  to  find 
the  true  remainder,  we  have  the  following 

Rule. 

To  the  first  remainder  add  the  products  which  arise  by 
multiplying  each  of  the  following  remainders  by  all  the  pre- 
ceding divisoxs,  except  its  own;  their  sum  will  be  the  true 
remainder. 

Examples. 

1.   Let  it  be  required  to  divide  43720  by  45  =  3x5x3. 
3)43720 

5  )  14573     .     1  =  1st  rem  .....          1 

3  )  2914     .     3  =  2d  rem.     .       .        3x3  =  9 

971     .     1  =  3d  rem.     .     1  X  5  X  3  =  15 

True  remainder,     25 

What  is  the  rule  for  finding  the  true  remainder? 


SIMPLE   NUMBERS.  65 


2.  Divide  956789  by  56. 

3.  Divide  4870029  by  72. 

4.  Divide  674201  by  110. 

5.  Divide  445767  by  144. 


6.  Divide  1913578  by  42. 

7.  Divide  146187  by  105. 

8.  Divide  26964  by  110. 

9.  Divide  93696  by  231. 


CASE    II. 
77.  When  the  divisor  is  10,  100,  1000,  &c. 

1.    In  476  yards  of  cloth,  how  many  pieces  are  there  of 
0  yards  each  ? 

ANALYSIS.  —  There  will  be  one-tenth  as       OPERATION. 
nany  pieces  as  there  are  yards;  47  tens        10  )  47  i  6 
is  one-tenth  of  47  hundreds  :  if,  then,  we  ^  .    , 

strike  off  the  right-hand  figure,  we  ob- 

tain one-tenth  of  476,  which  is  47,  and 

47T7r  quotient. 
6  over. 

If  the  divisor  is  100,  the  quotient  is  one-hundredth  of  the  divi- 
dend; 4  is  one-hundredth  of  4  hundreds:  if,  then,  we  strike  off 
the  two  right-hand  figures,  thus,  4|76,  we  obtain  one-hundredth 
of  476,  and  76  over.  Hence,  the  following 

Rule. 

I.  From  the  right  hand,  cut  off,  by  a  line,  as  many  fig- 
ures as  there  are  ciphers  in  the  divisor: 

II.  The  figures  at  the  left  will  be  the  quotient,  and  those 
at  the  right,  the  remainder. 

Examples. 


1.  Divide  49763  by  10.  * 

2.  Divide  7641200  by  100. 


3.  Divide  496321  by  1000. 

4.  Divide  64978  by  10000. 


CASE   .HI. 

78.    "When   the   divisor    contains    significant    figures,    -with 
ciphers  on  the  right  of  them. 

77.  What  is  the  rule  when  the  divisor  is  1,  with  any  number  o 
ciphers  annexed  ?  i  u £ 


()6  DIVISION. 

1.   Let  it  b9  required  to  divide  67389  by  700. 

ANALYSIS.— We    may    regard  OPERATION. 

the  divisor  as  a  composite  num-         f|00 }  673189 
ber,  of  which  the  factors  are  100 

and  7.     We  first  divide  by  100,  96  .  .  1  remains, 

by  striking  off  the  89,  and  then  189  true  remainder, 

by  7,  giving  90,  with  a  remain-  Ans.   961f§. 

der  of   1 ;    this   remainder   we 

multiply  by  the  first  divisor,  100,  and  then  add  89,  forming  the 
true  remainder,  189:  to  the  quotient,  96,  we  annex  189  divided 
by  700,  for  the  entire  quotient.  Hence,  the  following 

Rule. 

I.  Cut  off  the  ciphers  by  a  line,  and  cut  off  the  same 
number  of  figures  from  the  right  of  the  dividend. 

II.  Divide  the  remaining  figures  of  the  dividend  by  the 
remaining  figures  of  the  divisor,  and  find  the  true  remainder. 

III.  To  the  quotient  before  found,  annex  the  true  remain- 
der, divided  by  the  divisor,  for  the  true  quotient. 

Examples. 


1.  986327  by  210000. 

2.  876000  by  6000. 

3.  36599503  by  400700. 

4.  5714364900  by  36500. 

5.  18490700  by  73000. 


6.  7080^149  by  31500. 

7.  8749632  by  3700. 

8.  65975090  by  504000. 

9.  87470469  by  8510000. 
10.  97621397  by  6987000 


Applications. 

79.  Abstractly,  the  object  of  Division,  is  to  find  how  many 
times  one  number  is  contained  in  another.  Practically,  all 
questions  are  reduced  to  three  classes : 

1.  To  find   the  value  of  any  equal  part  of  a  number,  or 
of  any  number  of  things. 

2.  Knowing  the  entire  cost  of  a  number  of  things,  and 
the  price  of  a  single  thing,  to  find  the  number  of  things. 

78.  How  do  you  divide  when  the  divisor  is  any  digit,  with  ciphers 
on  the  right  of  it  ? 


APPLICATIONS.  67 

8.  Knowing  the  number  of  things,  and  their  entire  cost, 
to  find  the  price  of  a  single  thing. 

The  three  analyses  on  pages  54  and  55,  give  the  following 

Rules. 

1.   To  find  the  value  of  each  part. 

Divide  the  number  of  things  by  the  number  of  j)arls 
into  which  they  are  to  be  divided :  the  quotient  will  be  -the 
value  of  each  part. 

2.   To  find  the  number  of  things. 

Divide  the  entire  cost  by  the  price  of  a  single  thing :  the 
quotient  will  be  the  number  of  things. 

3.   To  find  the  price  of  a  single  thing. 
Divide  the  entire  cost  by  the  number  of  things :  the  quotient 
will  be  the  price  of  a  single  thing. 

Applications  in  the  previous  Rules. 

1.  Mr.  Jones  died,  leaving  an  estate  worth  4500  dollars, 
to  be  divided  equally  among  3  daughters  and  2  sons  :  what 
was  the  share  of  each  ? 

2.  The  owner  of  an  estate  sold  240  acres  of  land,  and 
had  312  acres  left :  how  many  acres  had  he  at  first  ? 

3.  The  sum  of  two  numbers  is  3475,  and  the  smaller  is 
1162:   what  is  the  greater? 

4.  The  difference  between  two  numbers  is  1475,   and  the 
greater  number  is  5760:  what  is  the  smaller? 

5.  A  gentleman  bought  a  house  for  two  thousand  twenty- 
five   dollars,    and    furnished   it    for    seven   hundred   and   six 
dollars ;  he  paid,  at  one  .time,  one  thousand  and  ten  dollars, 
and  at  another  time,  twelve  hundred  and  seven  dollars  :  how 
much  remained  unpaid  ? 

6.  At  a  certain  election,  the  whole  number  of  votes  cast 

79.  What  is  the  object  of  Division,  abstractly?  Practically,  to 
how  many  forms  are  all  questions  reduced  ?  What  are  they  ?  What 
are  the  rules  ? 


68  APPLICATIONS. 

for  two  opposing  candidates,  was  12672  ;  the  successful  can- 
didate received  316  majority:  how  many  votes  did  each 
receive  ? 

7.  What  number  must  be  multiplied  by  124,  to  produce 
40796? 

8.  The  sum  of  19125  dollars  is  to  be  distributed  equally 
among  a  certain  number  of  men,  each  to  receive  425  dollars : 
how  many  men  are  to  receive  the  money? 

9.  A   merchant   has   5100  pounds  of  tea,   and  wishes   to 
pack  it  in  60 'chests:  how  much  must  he  put  in  each  chest? 

10.  The  product  of  two  numbers  is  51679680,  and  one  of 
the  factors  is  615:   what  is  the  other  factor? 

11.  Bought    156   barrels    of  flour   for    1092   dollars,  and 
sold  the  same  for  9  dollars  per  barrel :  how  much  did  I  gain  ? 

12.  Mr.  James  has  14  calves,  worth  4  dollars  each  ;    4(^ 
sheep,  worth  3  dollars  each ;  he  gives  them  all  for  a  horse, 
worth  150  dollars  :  what  does  he  make  or  lose  by  the  bargain? 

13.  Mr.  Wilson  sells  4  tons  of  hay,  at  12  dollars  per  ton, 
80   bushels  of  wheat,   at   1   dollar  per  bushel,,  and  takes   in 
payment  a  horse,  worth  65  dollars,  a  wagon,  worth  40  dol- 
lars, and  the  rest  in  cash :   how  much  money  did  he  receive  ? 

14.  If  the   remainder   is    17,    the   quotient    610,    and   the 
dividend  45767,  what  is  the  divisor? 

15.  If  the  product  of  two  numbers  is  346712,  and  one  of 
the  factors  is  76,  what  is  the  other  factor? 

16.  If  the  quotient  is  482,  and  the  dividend  135442,  what 
is  the  divisor? 

17.  There  are  31173  verses  in  the  Bible  :  how  many  verses 
must  be  read  each  day,  that  it  may  be  read  through  in  a 
common  year  ? 

18.  The   distance    around   the   earth   is   computed    to   be 
about  25000  miles :  how  long  would  it  take  a  man  to  travel 
that  distance,   supposing   him   to   travel   at   the   rate   of  3£ 
miles  a  day  ? 

19.  A  person  having  a  yearly  salary  of  1500  dollars,  save.' 
at  the  end  of  the  year,  405  dollars :   what  were  his  average 
daily  expenses,  allowing  365  days  to  the  year? 


APPLICATIONS.  69 

20.  The  salary  of  the  President  of  the  United  States  1^ 
26000  dollars  a  year  :   how  much  r™   ue  spend  daily,   and 
save  of  his  salary,  iy*25  dollars  at  the  end  of  the  year? 

21.  A  speculator  bought   512   barrels   of  flour  for  3584 
dollars,  and  sold  the  same  for  4608  dollars :   how  much  did 
he  gain  per  barrel  ? 

22.  Mr.  Place  purchased  15  cows ;  he  sold  9  of  them  for 
35  dollars  apiece,  and  the  remainder  for  32  dollars  apieee, 
when  he  found  that  he  had  lost  123  dollars:  how  much  did 
he  pay  apiece  for  the  cows  ? 

23.  If  a  man's  salary  is   800  dollars  a  year,  and  his  ex- 
penses 425  dollars,  how  many  years  will  elapse  before  he  will 
be  worth  10000  dollars,  if  he  is  worth  2500  dollars  at  the 
present  time  ? 

24.  How  long  can  125  men  subsist  on  an  amount  of  food 
that  will  last  1  man  4500  days  ? 

25.  The  income  of  the  Bishop  of  Durham,  in  England,  is 
292  dollars  a  day :  how  many  clergymen  would  this  support 
on  a  salary  of  730  dollars  per  annum  ? 

26.  The   diameter   of  the   earth   is    7912   miles,  and   the 
diameter  of  the  sun,  112  times  as  great:  what  is  the  diam- 
eter of  the  sun  ? 

27.  By  the  census  of  1850,  the  whole  population  of  the 
United  States  was   23191876  ;   the  number  of  births  for  the 
previous    year    was    629444,    and    the    number    of,  deaths, 
324394  :    supposing  the  births  to   be  the  only  source  of  in- 
crease, what   was   the   population   at   the   beginning   of  the 
previous  year? 

28.  A  farmer  purchased  a  farm,  for  which  he  paid  18050 
dollars ;   he  sold  50   acres   for  60  dollars  an  acre,  and  the 
remainder  stood  him  in  50  dollars  an  acre :  how  much  land 
did  he  purchase  ? 

29.  A  merchant  bought  a  hogshead  of  molasses,  contain- 
ing 96  gallons,  at  35  cents  per  gallon ;  but  26 'gallons  leaked 
out,  and  he  sold  the  remainder  at  50  cents  psr  gallon:  did 
he  gain  or  lose,  and  how  much  ? 

80.    Mr.  James  bought  of  Mr.  Johnson  two  farms,  one  con- 


70  APPLICATIONS. 

taiuing  250  acres,  for  which  he  paid  85  dollars  per  acre; 
the  second  containing  1  ?5  acres,  for  which  he  paid  70  dollars 
an  acre ;  he  then  sold  them  both,  for  75  dollars  an  acre : 
did  he  make  or  lose,  and  how  much  ? 

31.  A  farmer  has  279  dollars,  with  which  he  wishes  to 
buy  cows  at   25  dollars,  sheep  at  4  dollars,  and  pigs  at  2 
dollars  apiece,  of  each  an  equal  number :   how  many  can  he 
buy  of  each  sort,? 

32.  Two  persons  counting  their  money,  found  that  together 
they  had   342   dollars ;    but  one   had   14  dollars   more   than 
one-half  of  it :    how  many  dollars  had  each  ? 

33.  Mr.  Bailey  has   7   calves,   worth  4   dollars   apiece,   9 
sheep,  worth  3  dollars  apiece,  and  a  fine  horse,  worth  175 
dollars.     He  exchanges  them  for  a  yoke  of  oxen,  worth  125 
dollars,  and  a  colt,  worth  65  dollars,  and  takes  the  balance 
in  hogs,  at  8  dollars  apiece:  how  many  hogs  does  he  take? 

34.  How  many  pounds  of  coffee,  worth  12  cents  a  pound, 
must   be   given  for  368  pounds   of  sugar,  worth  9  cents  a 
pound  ? 

35.  If  600  barrels  of  flour  cost  4800  dollars,  what  will 
2172  barrels  cost? 

36.  Mr.  Snooks,   the   tailor,   bought   of  Mr.   Squires,   the 
merchant,  4  pieces  of  cloth ;  the  first  and  second  pieces  each 
measured  45  yards,  the  third,  47  yards,  and  the  fourth,  53 
yards;  for  the  whole  he  paid  760  dollars:  what  did  he  pay 
for  35  yards? 

37.  What  is  the  difference  between  the  cost  of  a  flock  of 
sheep  containing  175,  at  4  dollars  apiece,  and  a  drove  of  97 
cattle,  at  85  dollars  apiece  ? 

38.  A  miller  bought  320  bushels  of  wheat  for  576  dollars, 
and  sold  256  bushels  for  480  dollars:  what  did  the  remain- 
der cost,  him  per  bushel? 

39.  A  merchant  bought  117  yards  of  cloth  for  702  dollars, 
and  sold  76  yards  of  it  at  the  same  price  for  which  he  bought 
it :   what  was  the  value  of  the  cloth  sold  ? 

40.  If  46   acres  of  land  produce   2484   bushels  of  corn, 
how  many  bushels  will  120  acres  produce? 


APPLICATIONS.  71 

41.  Mr.  Gill,  a  drover,  purchased   36  head  of  cattle,  at 
64  dollars  a  head,  and   88   sheep,  at  5  dollars   a  head :    he 
sold  the  cattle  for  40  dollars   a  head,  and  the  sheep  for  4 
dollars  apiece  :   did  he  make  or  lose,  and  how  much  ? 

42.  Mrs.  Louisa  Wilsie  has  3  houses,  valued  at  12530  dol- 
lars,  11324  dollars,  and  9875   dollars  :    also  a  farm,  worth 
6720  dollars.      She   has   a   daughter   and   2   sons.      To   the 
daughter  she  gives   one-third  the  value   of  the   houses   and 
one-fourth    the    value    of   the    farm,    and    then    divides    the 
remainder   equally   among    the    boys :    how   much   did    each 
receive  ? 

43.  Mr.  Jones  has  a  farm  of  250  acres,  worth  125  dollars 
per  acre,   and  offers'  to   exchange  with  Mr.  Gushing,  whose 
farm  contains  185  acres,  provided  Mr.  Gushing  will  pay  him 
20150    dollars    difference  :    what   was    Mr.    Cushing's    farm 
valued  at,  per  acre? 

44.  Mr.  Sparks  bought  a  third   part   of  neighbor  Spend- 
thrift's farm  for  2750  dollars  :  what  would  he  have  paid  for 
the  whole  farm  at  the  same  rate? 

45.  George  Wilson  bought  24  barrels  of  pork,  at  14  dol- 
lars a  barrel ;  one-fourth  of  it  proved  damaged,  and  he  sold 
it  at  half  price,  and  the  remainder  he  sold  at  an  advance  of 
3  dollars  a  barrel :   did  he  make  or  lose  by  the   operation, 
and  how  much? 

46.  A  gentleman,  having  50000  dollars,  spent  half  of  it 
in  buying  5  houses,  which,  after  repairing  at  an  expense   of 
1250  dollars,  he  sold  at  6520   dollars   each  :    what  was  his 
fortune  after  the  transaction  ? 

47.  A  gentleman  bought  3  houses  for  15850  dollars.    For 
two  he  paid  an  equal  price  ;    and  for  the  third,  850  dollars 
more  than  for  either  of  the   others  :   what  did   he   pay  for 
each? 

48.  Mr.  J.  Williams  went  into  business  with  a  capital  of 
25000  dollars  :  in  the  first  year  he  gained  2000  ;  in  the  second 
year,  3500 ;  in  the  third  year,  4000  dollars :  he  then  invested 
the  whole  in  a  cargo  of  tea  and  doubled  his  money:   what 
was  then  the  value  of  his  fortune  ? 


PROPERTIES   OF  NUMBERS. 


PROPERTIES  OF  NUMBERS. 

Exact  Divisors. 

80.  An  EXACT  DIVISOR  of  a  number,  is  any  number,  ex- 
cept 1  and  the  number  itself,  that  will  divide  it  without  a 
remainder.     The  dividend  is  then  said  to  be  divisible  by  the 
divisor. 

81.  An  ODD  NUMBER    is  not  divisible  by  2. 

82.  An  EVEN  NUMBER   is  one  divisible  by  2. 

1.  Three,  is  an  exact  divisor  of  any 'number,  the  sum  of 
whose  digits  is  divisible  by  3. 

2.  Four,  is  an  exact  divisor  of  a  number,  when  it  will  exact- 
ly divide  the  number  expressed  by  the  two  right-hand  digits. 

3.  Five,  is  an  exact  divisor  of  every  number  whose  right- 
hand  figure  is  0  or  5. 

4.  Six,  is  an  exact  divisor  of  any  even  number  of  which 
3  is  an  exact  divisor. 

5.  Nine,  is  an  exact  divisor  of  any  number,  the  sum  of 
whose  digits  is  divisible  by -it. 

6.  Ten,  is  an  exact  divisor  of  every  number  whose  right- 
hand  figure  is  0. 

83.  A  PRIME  NUMBER  is  one  which  has  no  exact  divisor: 

1,     2,     3,     5,     7,     11,     13,     17,     19,  &c., 
are  prime  numbers. 

84.  A  COMPOSITE  NUMBER  is  a  number  which  has  two  or 
more  exact  divisors. 

85.  A  FACTOR  of  a  composite  number,  is  any  one  of  its 
exact  divisors. 

80.  What  is  an  exact  divisor  of  any  number?  What  is  then 
said  of  "the  dividend?— 81.  What  is  an  odd  number?— 82.  What  is 
an  even  number? — 83.  What  is  a  prime  number? — 84.  What  is  a 
composite  number? — 85.  What  is  a  factor? 


PRIME   FACTORS.  73 

C  A  S  E     I . 

86.    To  find  the  prime  factors  of  a  composite  number. 
1.  What  are  the  prime  factors  of  2310? 

OPERATION 

ANALYSIS. — We  first  divide  by  2,  the  least  prime        2  )  2310 
factor  of  the  given  number.     We  then  divide  the  . 

quotient  by  3,  then  the  quotient  by  5,  and  then  by 
V,  when  we  obtain  the  quotient  11,  which  is  prime.        5  )    385 
Hence,  the  prime  factors  of  2310  are,  2,  8,  5,  7,        *  v ** 
and  11.     Hence,  the  following 

11 
Rule. 

Divide  the  given  number  by  any  prime  number  that  will 
exactly  divide  it :  then  divide  the  quotient  in  the  same 
manner,  and  so  on,  till  a  quotient  is  found  which  is  a  prime 
number:  the  several  divisors  and  the  last  quotient  will  be 
the  prime  factors. 

Examples. 

What  are  the  prime  factors  of  the  following  numbers  ? 


1.  Of  the  number  9  ? 

2.  Of  the  number  15  ? 


6.  Of  the  number  32  ? 

7.  Of  the  number  48? 


3.  Of  the  number  24  ?  8.    Of  the  number  56  ? 

4.  Of  the  number  16?  9.    Of  the  number  63? 

5.  Of  the  number  18  ?  10.   Of  the  number  76  ? 

CASE    II. 

87.    To  find  the  prime  factors   common  to  two  or  more 
composite  numbers. 

1.  What  are  the  common  prime  factors  of  70,  210,  and 
280? 

ANALYSIS. — It  is   plain  that  2  is  an  OPERATION. 
exact  divisor  of  all  the  numbers,   and 

hence,  a  common  factor :  5  is  an  exact  -     2  )  «0  •  2I°  •  ^80 

divisor  of  the  first  set  of  quotients,  85,  5  )  35  .   105  .  140 

105,  and  140 ;   hence,  it  is  a  common  ^  7    ^       ^ ^ 

factor :  7  is  an  exact  divisor  of  the  sec-  ^ -— 

ond  set  of  quotients ;  hence,  it  is  a  com-  134 
4 


74  PROPERTIES   OF   NUMBERS. 

mon  factor,  and  the  third  set  of  quotients  have  no  exact  divisor. 
Hence,  the  following 

Rule. 

I.  Write  the  numbers  in  a  row,  and  then  divide  them 
by  any  prime  number  that  is  an  exact  divisor  of  all  of 
them  : 

II.  Divide  each  set  of  quotients  in  the  same  manner, 
until  a  set  is  found  which   has    no   exact  divisor.     The 
divisors  will  be  the  common  prime  factors. 

NOTE. — The  product  of  the  prime  factors,  is  the  greatest  factor 
common  to  all  the  numbers.  Thus,  2x5x7  =  70,  is  the 
greatest  factor  common  to  70,  210,  280. 

Examples. 

1.  What  are  the  prime  factors  common  to  6,  9,  and  24  ? 

2.  What  are  the  prime  factors  common  to  21,  63,  and  84  ? 

3.  What  are  the  prime  factors  common  to  21,  63,  and  105  ? 

4.  What  are  the  common  prime  factors  of  28,  42,  and  10  ? 

5.  What  are  the  prime  common  factors  of  84,  126,  and  210  ? 

6.  What  are  the  prime  factors  of  210,  315,  and  525  ?' 

Cancellation. 

88.  CANCELLATION  is  a  process  of  shortening  Arithmetical 
operations    in    Division,   by   omitting,    or   canceling,   factors 
common  to  the  dividend  and  divisor. 

89.  Cancellation  depends  upon  the  principle  that, 

If  the  dividend  and  divisor  be  both  divided  by  the  same 
number,  the  quotient  will  not  be  changed. 

86.  How  do  you  find  the  factors  of  a  composite  number? 

87.  How  do  you  find  the  prime  factors  common  to  two  or-  more 
composite  numbers  ?    What  is  the  greatest  factor  common  to  aU 
of  the  numbers  ? 

88.  What  is  Cancellation  ? 

89.  On  what  principle  does  Cancellation  depend? 


CANCELLATION.  76 

1.  Divide  63  by  21. 

ANALYSIS. — Resolve  the  dividend  and  di-  OPERATION. 

visor  into  factors,  then  cancel  those  which      63  _  H  X  9  _  „ 
are    common,   and   mark   the    canceled  fig-  ~~  m     0  ~~ 

&L         /I  X  o 

ures. 

2.  In  7  times  56,  how  many  times  8  ? 
ANALYSIS.— Resolve   56  into  the  OPERATION. 

two  factors  7  and  8,  and  then  can-       56  X  7  _  0x7x7  _ 
eel  the  8.  8  $ 

3.  In  36  times  15,  how  many  times  45  ? 

ANALYSIS. — We  see  that  9  is  a  factor  of  36 

and  45.     Divide  by  this  factor,  and  write  the  OPERATION. 

quotient  4  over  36,  and  the  quotient  5  below  4.        3 

45.     Again,  5  is  a  factor  of  15  and  5.     Divide  £0  ^  %fy       12 

15  by  5,  and  write   the   quotient  3  over  15,  — ->£t —  ~  ~T* 
and  the  quotient  of  5  by  5,  under  o.    Dividing 
5  by  5,  reduces  the  divisor  to  1 :   hence,  the 

true  quotient  is,  -   —  =  —  =  12. 

90.  Hence,  for  the  operations  of  Cancellation,  we  have 
the  following 

Rule. 

Cancel  those  factors  that  are  common  to  the  dividend 
and  divisor,  and  then  divide  the  product  of  the  remaining 
factors  of  the  dividend  by  the  product  of  the  remaining 
factors  of  the  divisor. 

NOTES. — 1.  If  one  of  the  numbers  contains  a  factor  equal  to 
the  product  of  two  or  more  factors  of  the  other,  they  may  all  be 
canceled. 

2.  If  the  product  of  two   or  more  factors  of  the  dividend  is 
equal  to  the  product  of  two  or  more  factors  of  the  divisor,  they 
may  all  be.  canceled. 

3.  If  all  the  factors  of  the  dividend  are  canceled,  the  quotient 
I  must  be  put  for  the  factor  last  canceled. 

90.  What  is  the  rule  for  the  operations  of  Cancellation? 


76  PROPERTIES   OF   NUMBERS.' 

Examples. 

1.  What  number  is  equal  to  the  product  of  36  and  13, 
divided  by  the  product  of  4  and  9  ? 

ANALYSIS.— We  see  that  4  times  9  OPERATION. 

are  equal  to  36;  therefore,  we  cancel          30  X  13  __  13  ,~ 

the  36,  and  the  4  and  9.  ^X0  1 

2.  Divide  960  by  480. 

ANALYSIS. — We  see  that  10  is  a  com-  OPERATION. 

mon  factor;  then  12,  then  4.    We  may  959        96       8 

divide  mentally,  by  the  common  fac-  —  =  —  =  —  =  2. 
tors,  and  place  the  results  at  the  right. 

3.  Divide  the   product   of  6x7x9x11,   by  2x3x7x3 
X21. 

4.  Divide  the   product   of  4  x  14  x  16  x  24,   by   7x8x32 
X12. 

5.  Divide  the   product  of  5x11x9x7x15x6,    by   30 
y  3x21x3x5. 

6.  Divide  285120  by  5184. 

7.  Divide  5080320  by  635040. 

8.  How  much  calico,  at  25  cents  a  yard,  must  be  given 
for  8700  cents  ? 

.  9.    How  many  yards  of  cloth,  at  46  cents  a  yard,  can  be 
bought  for  2116  cents  ? 

10.  How  much   molasses,   at  42  cents   a   gallon,   can  be 
bought  for  1512  cents? 

11.  In  a  certain  operation,  the  numbers  24,  28,  32,  49, 
81,  are  to   be  multiplied  together,  and  the  product  divided 
by  8x4x7x9x6:    what  is  the  result  ? 

12.  How  many  pounds  of  butter,  worth  15  cents  a  pound, 
may  be  bought  for  25  pounds  of  tea,  at  48  cents  a  pound? 

13.  How  many  bushels  of  oats,  at  42  cents  a  bushel,  must 
be  given  for  3  boxes  of  raisins,  each  containing  26  pounds, 
at  14  cents  a  pound  ? 

14.  A  man  buys  2  pieces  of  cotton  cloth,  each  containing 
33  yards,  at  11  cents  a  yard,  and  pays  for  it  in  butter  at  18 
cents  a  pound  :   how  many  pounds  of  butter  did  ho  give  ? 


LEAST   COMMON   MULTIPLE.  77 

15.  If  sugar  can  be  bought  for  7  cents   a  pound,  how 
many  bushels  of  oats,  at  42  cents  a  bushel,  must  I  give  for 
56  pounds  ? 

16.  Bought  48  yards  of  cloth,  at  125  cents  a  yard  :  bow- 
many  bushels  of  potatoes  are  required  to  pay  for  it,  at  150 
cents  a  bushel  ? 

17.  Mr.  Butcher  sold  342  pounds  of  beef,  at  6  cents  a 
pound,  and  received  his  pay  in  molasses  at  36  cents  a  gallon : 
how  many  gallons  did  he  receive  ? 

18.  Mr.  Farmer  sold  1263  pounds  of  wool,  at  5  cents  a 
pound,  and  took  his  pay  in  cloth  at  441  cents  a  yard :  how 
many  yards  did  he  take  ? 

19.  How  many  firkins  of  butter,  each  containing  56  pounds, 
at  18  cents  a  pound,  must  be  given  for  3  barrels  of  sugar, 
each  containing  200  pounds,  at  9  cents  a  pound  ? 

20.  How  many  boxes  of  tea,  each  containing  24  pounds, 
worth  5  shillings  a  pound,  must  be  given  for  4  bins  of  wheat, 
each  containing  145  bushels,  at  12  shillings  a  bushel? 

21.  A.  worked  for  B.  8  days,  at  6  shillings   a  day,   for 
which  he  received  12  bushels  of  corn  :   how  much  was  the 
corn  worth  a  bushel  ? 

22.  Bought  15  barrels  of  apples,  each  containing  2  bush- 
els, at  the  rate  of  3.  shillings  a  bushel :   how  many  cheeses, 
each  weighing  30  pounds,  at  1  shilling  a  pound,  will  pay  for 
the  apples  ? 

Least  Common  Multiple. 

91.  A  MULTIPLE  of  a  number,  is  the  product  of  that  num- 
ber by  some  other  number.     Thus,  the  dividend  is  a  multiple 
of  the  divisor  or  quotient. 

92.  A  COMMON  MULTIPLE  of  two  or  more  numbers,  is  a 
number  exactly  divisible  by  each  of  them. 

93.  The    LEAST    COMMON    MULTIPLE     of    two     or    more 
numbers,   is    the    least    number  which   is    divisible   by   each 

91.  What  is  a  multiple  of  a  number? 

92.  What  is  a  common  multiple  of  two  or  more  numbers  ? 

93.  What  is  the  least  common  multiple  of  two  or  more  numbers  ? 


78  PROPERTIES   OF   NUMBERS. 

of  them.     Thus,   18  is  the   least  common   multiple   of  2,  6, 
and  9. 

NOTES. — 1.  If  a  division  is  exact,  the  dividend  may  be  resolved 
into  two  factors,  one  of  which  will  be  the  divisor,  and  the  other 
the  quotient. 

2.  If  the  divisor  be  resolved  into  its  prime  factors,  the  corre- 
sponding factor  of  the,  dividend  may  be  resolved  into  the  same 
factors:  hence,  the  dividend  will  contain  every  factor  of  an  exact 
divisor. 

3.  The  question  of  finding  the  least  common  multiple  of  several 
numbers,  is,  therefore,  reduced  to  finding  a  number  which  shall 
contain  all  the  prime  factors  of  the  given  numbers,  and  none 
others. 

1.  Find  the  prime  factors  and  least  common  multiple  of 
6,  12,  and  18. 

ANALYSIS. — "Write  tbe  numbers  in  a  line,  OPEKATTON. 

and  then   divide   them    and   the   quotients         2)6  .  12  .   18 
which  follow,    by  any  prime  number  that  ~ 

will  exactly  divide  two  or  more   of  them,  ' 

until  quotients  are  found  which  are  prime  123 

with  each  other.     It  is  plain,  that  the  divi- 
sors, 2  and  3,  are  prime  factors  of  6;   2,  3,  and  the  quotient  2, 
of  12;   and  2,  3,  and  the  quotient  3,  of  18:   hence,  the   prime 
factors  are  2,  3,  2,  and  3,  and  their  product,    2x3x2x3  =  3  6, 
the  least  common  multiple. 

94.   Hence,  to  find  the  least  common  multiple, 

Rule. 

I.  Write  the  numbers  in  a  line,  and  divide  by  any  prime 
number  that  will  exactly  divide  any  two  of  them,  and  write 
down  the  quotients,  and  the  undivided  numbers. 

II.  Divide  as  before,  until  there  is  no  exact  divisor  of 
any  two  of  the  quotients :  the  product  of  the  divisors  and 
the  final  quotients,  will  be  the  least  common  multiple. 

Examples. 

1.  Find  the  least  common  multiple  of  3,  4,  and  8. 

2.  Find  the  least  common  multiple  of  3,  8,  and  9. 


GREATEST   COMMON   DIVISOR.  79 

3.  Find  the  least  common  multiple  of  6,  7,  8,  and  10. 

4.  Find  the  least  common  multiple  of  21  and  49. 

5.  Find  the  least  common  multiple  of  2,  7,  5,  6,  and  8. 

6.  Find  the  least  common  multiple  of  4,  14,  28,  and  98. 

7.  Find  the  least  common  multiple  of  13  and  6. 

8.  Find  the  least  common  multiple  of  12,  4,  and  7. 

9.  Find  the  least  common  multiple  of  6,  9,  4,  14,  and  16 
10.  Find  the  least  common  multiple  of  13,  12,  and  4. 


Greatest  Common  Divisor. 

95.  A  COMMON  DIVISOR  of  two  or  more  numbers,  is  any 
number  that  will  divide  each  of  them  without  a  remainder. 

96.  The  GREATEST  COMMON  DIVISOR  of  two  or  more  num- 
bers, is  the  greatest  number  that  will  divide  each  of  them 
without  a  remainder. 

97.  Two  numbers  are  prime  with  each  other,  when  they 
have  no  common  divisor. 


CASE    I. 

98.  To  find  the  greatest  common  divisor  of  two  or  more 
numbers,  when  the  numbers  are  small. 

Since  an  exact  divisor  is  a  factor,  the  greatest  common 
divisor  of  the  given  numbers,  will  be  their  greatest  common 
factor :  Hence, 

Rule. 

Find  the  prime  factors  common  to  all  the  numbers 
(Art.  87),  and  their  product  will  be  the  greatest  common 
divisor. 

94.  What  is  the  rule  for  finding  the  least  common  multiple? 

95.  What  is  a  common  divisor  of  two  or  more  numbers  ? 

96.  What  is  the  greatest  common  divisor  of  two  or  more  numbers  ? 

97.  When  are  two  numbers  prime  with  each  other? 

98.  How  do  you  find  the  greatest  common  divisor,  when  the 
numbers  are  small  ? 


80  PROPERTIES   OF    NUMBERS. 

Examples. 

1.  What  is  the  greatest  common  divisor  of  24  and  30  ? 

2.  What  is  the  greatest  common  divisor  of  9  and  18  ? 

3.  What  is  the  greatest  common  divisor  of  6,  12,  and  30  ? 

4.  What  is  the  greatest  common  divisor  of  15,  25,  and  30  ? 

5.  What  is  the  greatest  common  divisor  of  12,  18,  and  72? 

6.  What  is  the  greatest  common  divisor  of  25,  35,  and  70  ? 

7.  What  is  the  greatest  common  divisor  of  28,  42,  and  70  ? 

8.  What  is  the  greatest  common  divisor  of  84, 126,  and  210  ? 

CASE    II. 

99.    To  find  the  greatest  common  divisor,  when  the  num- 
bers are  large. 

The  operation  of  finding  the  common  divisor,  depends  on 
the  following  principles : 

1.  Any  number  which  will  exactly  divide       ILLUSTRATION. 
the  difference  of  two  numbers,   and  one  of        24  —  16  =  8 
them,  will  exactly  divide  the  other ;  else,  we 

should  have  a  whole  number  equal  to  a  fraction,  which  is 
impossible. 

2.  Any  number  that  will  exactly  divide  another,  will  divide 
any  multiple  of  that  other;   because,  the  first  dividend  is  a 
factor  of  the  multiple,  and  any  number  which  will  divide  a 
factor,  will  divide  the  multiple. 

1.  What  is  the  greatest  common  divisor  of  25  and  70  ? 

ANALYSIS.— Divide  the  greater  mim-  OPERATION 

ber,   TO,   by  the   less,   25;   we   find   a  ( 

quotient  2,  and  a  remainder  20.     Then  50 

divide  the  divisor  25  by  the  remainder  

20 ;  the  quotient  is  1,  and  the  remain-  20  )  25  (  1 

der  5.     Then  divide  the  divisor  20  by  20 

the    remainder  5 ;    the   quotient   is  4,  ^\  go  (  4 

and  the  division  exact.  20 

Now,  the  remainder  5,  exactly  divides 
itself  and  20 ;  hence,  by  the  first  prin- 
ciple, it  will  exactly  divide  25.     Since  5  divides  25,  it  will,  by 
the   second  principle,  divide  50,  a  multiple  of  25;   but  since  it 


GREATEST   COMMON    DIVISOR.  81 

divides  the  difference,  20,  and  one  number,  50,  it  will  divide  70: 
hence,  it  is  a  common  divisor  of  25  and  70;  and  since  there  is 
no  other  common  factor,  it  is  the  greatest  common  divisor. 

Hence,  to  find  the  greatest  common  divisor, 

Rule. 

Divide  the  greater  number  by  the  less,  and  then  divide 
the  preceding  divisor  by  the  remainder,  and  so  on,  till 
nothing  remains:  the  last  divisor  will  be  the  greatest  com- 
mon divisor. 

Examples. 

1.  What  is  the  greatest  common  divisor  of  216  and  408? 

2.  Find  the  greatest  common  divisor  of  408  and  740. 

3.  Find  the  greatest  common  divisor  of  315  and  810. 

4.  Find  the  greatest  common  divisor  of  4410  and  5670. 

5.  Find  the  greatest  common  divisor  of  3471  and  1869. 

6.  Find  the  greatest  common  divisor  of  1584  and  2772. 

NOTE. — If  it  he  required  to  find  the  greatest  common  divisor 
of  more  than  two  numbers,  first  find  the  greatest  common  divi- 
sor of  two  of  them,  then  of  that  common  divisor  and  one  of  the 
remaining  numbers,  and  so  on  for  all  the  numbers:  the  last 
common  divisor  will  be  the  greatest  common  divisor  of  all  the 
numbers. 

7  What  is  the  greatest  common  divisor  of  492,  744,  and 
1044? 

8  What  is   the   greatest   common   divisor  of  944,   1488, 
and  2088  ? 

9.  What  is  the  greatest  common  divisor  of  216,  408,  and 
740  ? 

10.  What  is  the   greatest  common  divisor  of  945,   1560, 
and  22683? 


99.  How  do  you  find  the  greatest  common  divisor,  when  the 
numbers  are  large? 

4* 


82  COMMON   FRACTIONS. 


COMMON    FRACTIONS. 

100.  A  UNIT    is  a  single  thing  ;   as,  1  apple,  1  chairy  1 
pound  of  tea ;  and  is  denoted  by  1. 

If  a  unit  be  divided  into  two  equal  parts,  p^ch  part  is 
called,  one-half. 

If  a  unit  be  divided  into  three  equal  parts,  each  part  is 
called,  one-third. 

If  a  unit  be  divided  into  four  equal  parts,  each  part  is 
called,  one-fourth. 

If  a  unit  be  divided  into  twelve  equal  parts,  each  part  is 
called,  one-twelfth;  and  if  it  be  divided  into  any  number  of 
equal  parts,  we  have  a  like  expression  for  each  part. 

The  parts  are  thus  written : 

J    is  read,    one-half.  \  is  read,  one-seventh. 

J     .     .     .     one-third.  -J-  .     .     .  one-eighth. 

•J-     .     .     .     one-fourth.  iV  •     •     •  one-tenth. 

-J     .     .     .     one-fifth.  iV  •     •     •  one-fifteenth. 

^     .     .     .     one-sixth.  sV  •     •     •  one-fiftieth. 

101.  The  UNIT  OF  A  FRACTION   is  the  single  thing  that  is 
divided  into  equal  parts. 

102.  A  FRACTIONAL  UNIT  is  one  of  the  equal  parts  of  the 
unit  that  is  divided. 

NOTE. — In  every  fraction,  let  the  pupil  distinguish  carefully 
between  the  unit  of  the  fraction  and  the  fractional  unit.  The 
first  is  the  whole  thing  from  which  the  fractions  are  derived; 
the  second,  one  of  the  equal  parts  into  which  that  thing  is  divided. 

100.  What  is  a  unit?    By  what  is  it  denoted?    What  is  one- 
half?    One-third?    One-fourth?    One-twelfth? 

101.  What  is  the  unit  of  a  fraction  ? 

102.  What  is  a  fractional  unit?    What  is  the  difference  between 
the  unit  of  a  fraction  and  a  fractional  unit  ? 


COMMON   FRACTIONS.  83 

103.  Every  whole  number,  except  1,  has  a  fractional  unit 
corresponding  to  it :  thus,  the  numbers 

2,     3,     4,     5,     6,     7,     8,     9,     10,  &c., 
have,  corresponding  to  them,  the  fractional  units 
i    I    i,    J-    i,    *,    i    i,    T'S,  &c. 

If  we  suppose  a  class  of  boys  each  to  have  an  apple,  and 
that  the  apple  of  each  be  divided  into  equal  parts  corre- 
sponding to  his  class  number,  the  first  boy  will  have  the 
whole  apple,  or  the  unit  of  the  fraction  ;  the  second  boy  will 
have  the  whole  apple  in  the  two  fractional  units,  one-half ;  the 
third,  in  the  three  fractional  units,  one-third;  the  fourth,  in 
the  four  fractional  units,  one-fourth ;  and  each  boy  of  a 
higher  number,  will  have  the  whole  apple  in  as  many  frac- 
tional units  as  are  denoted  by  his  number  in  the  class. 

The  fractional  units  of  the  fourth  boy  may  be  derived  from 
those  of  the  second,  by  dividing  each  half  by  2,  giving  4 
fourths ;  the  units  of  the  6th  boy  may  be  derived  from  those 
of  the  2d,  by  dividing  each  by.  3,  or  from  those  of  the  3d, 
by  dividing  each  by  2 ;  and  similarly  for  any  of  the  higher 
numbers  which  are  multiples  of  the  lower. 

104.  An  INTEGRAL    or  WHOLE  NUMBER    is  the  unit  1,  or 
a  collection  of  units  1. 

105.  A  FRACTION   is  a  fractional  unit,  or  a  collection  of 
fractional  units. 


103.  What  is  the  fractional  unit  corresponding  to  2?    To  4? 
To  6  ?    To  12  ?    To  65  ?    If  each  of  a  class  of  boys  has  an  apple 
divided  into  parts  corresponding  to  his  number,  what  will  be  the 
fractional  unit  of  the  4th  boy  ?    How  many  fractional  units  will  he 
have?    How  may  they  be  derived  from  those  of  the  second  boy? 
What  will  be  the  fractional  unit  of  the  12th  boy  ?    From  those  of 
what  other  boys  may  they  be  derived  ?    How  from  the  24  ?    How 
from  the  3d?    How  from  the  4th?    How  from  the  6th? 

104.  What  is  an  integral,  or  whole  number? 

105.  What  is  a  fraction? 


84  COMMON   FRACTIONS. 

106.  Any  collection  of  fractional  units,  is  thus  written : 

•|      which  is  read,     2  halves       —    -3-  X  2. 

f  "          "  2  thirds       =    J  x  2. 

J  "          "  3  fourths     =1x3. 

|  "          "  4  fifths         =    j-  X  4. 

f  "          "  5  eighths     =    J  x  5. 

T7T          "          "  7  elevenths  =  ^  x  7. 

JJ          "          "         12  fifteenths  =  T^  x  12. 
&c.,  &c.f  &c.,  &c. 

Hence,  we  see  that  every  fraction  may  be  divided  into 
two  factors ;  one  of  which  is  the  fractional  unit,  and  the 
other,  the  number  denoting  how  many  times  the  fractional 
unit  is  taken. 

107.  The  DENOMINATOR   is  the  number  written  below  the 
line,  and  shows  into  how  many  equal  parts  the  unit  of  the 
fraction  is  divided. 

108.  The  NUMERATOR  is  the  number  written  above  the  line, 
and  shows  how  many  fractional  units  are  taken. 

109.  The  TERMS  of  a  fraction   are  the  numerator  and  de- 
nominator,   taken  together ;    hence,   every  fraction  has   two 
terms. 

110.  The  VALUE  of  a   fraction    is  the  number  of  times 
which  it  contains  the  unit  1. 

111.  To  ANALYZE  a  fraction    consists  in  naming  its  unit, 
its  fractional  unit,  and  the  number  of  fractional  units  taken  : 
Thus,  in  the  fraction  f,  the  unit  of  the  fraction  is  1 ;   the 
fractional  unit,  -J-;  and  the  number  of  fractional  units  taken 
is  3. 

106.  Explain  the  manner  of  writing  fractional  units.    Into  how 
many  factors  may  every  fraction  be  divided?    What  are  they? 

107.  What  is  the  denominator?    What  does  it  show? 

108.  What  is  the  numerator?    What  does  it  show? 

109.  What  are  the  terms  of  a  fraction  ?    How  many  terms  has 
every  fraction? 

110.  What  is  the  value  of  a  fraction  ? 

111.  What  is  the  analysis  of  a  fraction  ? 


PROPERTIES   OF   FRACTIONS.  85 

112.   A  whole  number  may  be  expressed  fractionally,  by 
writing  1  under  it  for  a  denominator.     Thus, 

3  may  be  written  ^  and  is  read,  3  ones. 

5  ....  f       ...  5  ones. 

6  ....  f       ...  6  ones. 
8     ....  f       ...  8  ones. 

113.    Properties  of  Fractions. 

1.  All  the  parts  of  the  unit  1,  however  divided,  make  up 
the  unit  itself;   hence,  any  fractional  unit,  multiplied  by  the 
number  of  parts,  is  equal  to  1. 

2.  If  the  numerator  is  less  than  the  number  of  parts,  the 
value  of  the  fraction  is  less  than  1. 

3.  If  the  numerator  is  greater  than  the  number  of  parts, 
some  of  the  fractional  units  must  have  come  from  a  second 
unit;   and  hence,   the  value  of  the   fraction  will  be  greater 
than  1. 

* 

Examples  in  writing  and  reading  Fractions. 

1.  Analyze  the  following  fractions: 

T5s>    *,    ¥.    A,    i     5%>    T¥T 

2.  Write  12  of  the  It  equal  parts  of  1. 

3.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit 
one-twentieth,   express   6   fractional  units;   express,  also,  12 
and  18. 

4.  If  the  fractional  unit  is  one  36th,  express  32  fractional 
units;  also,  35,  38,  54,  6,  8. 

5.  If  the  fractional  unit  is  one-fortieth,  express  9  fractional 
units;  also,  16,  25,  69,  75. 

6.  Write  forty-nine,  one  hundred  and  fifteenths. 

7.  Write  three  hundred  and  sixty-one,  forty-sevenths. 

112.  How  may  a  whole  number  be  expressed  fractionally? 

113.  When  is  a  fraction  equal  to  1  ?    When  less  than  1  ?    Whei 
greater  than  1? 


86  COMMON   FRACTIONS. 

8.  Write    seven   thousand    six   hundred   and   fifteen,   nine 
hundred  and  fifteenths. 

9.  Write  six  thousand  four  hundred,  elevenths. 

10.  Write  six  thousand  two  hundred  and  forty-two,  three 
hundred  and  fifty-thirds. 

Analyze  each  of  the  above  fractions,  when  written. 

114.    There  are  six  kinds  of  fractions: 

1.  A  PROPER   FRACTION    is   one  whose   numerator  is   less 
than  the  denominator. 

The  following  are  proper  fractions : 

*.   *.   i   *.   f   f    A,    I,    I- 

2.  An  IMPROPER  FRACTION  is  one  whose  numerator  is  equal 
to,  or  exceeds  the  denominator. 

The  following  are  improper  fractions: 

i,   I.   I-   I.   *,.  I   -¥-,   ¥,    !f- 

NOTE. — Such  a  fraction  is  called  improper,  because  its  value 
equals  or  exceeds  1. 

3.  A  SIMPLE  FRACTION    is  one  whose  numerator  and  de- 
nominator are  both  whole  numbers. 

The  following  are  simple  fractions : 

1  35  8  9          8  6          7 

T>        2^       "£>       T»       ?»       3>        3>       7- 

NOTE. — A  simple  fraction  may  be  either  proper  or  improper. 

4.  A  COMPOUND  FRACTION    is  a  fraction  of  a  fraction,  or 
several  fractions  connected  by  the  word  of,  or  x. 

The  following  are  compound  fractions : 

l  of  J,    J  of  i  of  i     Jx3,     *  x  J  X  4. 

5.  A  MIXED  NUMBER    is  the  sum  of  a  whole  number  and 
a  fraction. 

The  following  are  mixed  numbers : 

3J,       4},       6f,       5|,       6f,       3f 
114.  How  many  kinds  of  fractions  are  there  ?    Name  them. 


FUNDAMENTAL   PROPOSITIONS.  87 

6.    A  COMPLEX  FRACTION    is  one  whose  numerator  or  de- 
nominator is  fractional;  or,  in  which  both  are  fractional. 
The  following  are  complex  fractions : 

19V  f  6H' 

i 
Fundamental  Propositions. 

115.    Let  it  be  required  to  multiply  £  by  3. 

ANALYSIS. — In  £  there  are  5  fractional  OPERATION. 

units,  each  of  which  is  -£,  and  these  are         |-  X  3  =  >5"^  =  *£-. 
to  be  taken  3  times.     But  5  things  taken 

3  times,  gives  15  things  of  the  same  kind;   that  is,  15  sixths: 
hence, 

PROPOSITION  I. — If  the  numerator  of  a  fraction  be  multi- 
plied by  any  number,  the  value  of  the  fraction  ivill  be  in- 
creased as  many  times  as  there  are  units  in  the  multiplier. 


Examples. 


1.  Multiply  |  by  8. 

2.  Multiply  J  by  5. 

3.  Multiply  |  by  9. 


4.  Multiply  T89  by  14. 

5.  Multiply    J    by  20. 

6.  Multiply -yy7- by  25. 


116.   Let  it  be  required  to  multiply  %  by  3. 

ANALYSIS. — In  f  there  are  4  fractional  OPERATION. 

units,  each  of  which  is  |.    If  we  divide        £  x  3  =     *.  =  ^. 
the  denominator  by  3,  we  change  the 

fractional  unit  from  £  to  -£,  which  is  3  times  as  great  as  £.  If 
we  take  this  fractional  unit,  4  times,  is  before,  the  result,  4}  is  3 
times  as  great  as  |:  therefore,  we  have 

PROPOSITION  II. — If  the  denominator  of  a  fraction  be 
divided  by  any  number,  the  value  of  the  fraction  will  be  in- 
creased as  many  times  as  there  are  units  in  the  divisor. 


115.  What  is  Proposition  I.  ?— 116.  What  is  Proposition  II.  ? 


88  COMMON   FRACTIONS. 

Hence,  to  multiply  a  fraction,  divide  its  denominator. 
Examples. 


1.  Multiply  f  by  2,  by  4. 

2.  Multiply  if  by  2,  4,  8. 

3.  Multiply  ^V  by  2>  4>  6- 


4.  Multiply  if  by  2,  4,  6. 

5.  Multiply  JJ  by  2,  6,  7. 

6.  Multiply  £jft  by  5,  10. 


117.   Let  it  be  required  to  divide  T9r  by  3. 

ANALYSIS. — In  T9T  there  are  9  frac-  OPERATION. 

tional  units,  each  of  which  is  Ty,  and  9  _i_  g  _  9  -7-3  __  3  ^ 
these  are  to  be  divided  by  3.  But  9 

things,  divided  by  3,  gives  3  things  of  the  same  kind  for  a  quo- 
tient; hence,  the  quotient  is  3  elevenths,  a  number  one-third  as 
great  as  T9T:  hence,  we  have 

PROPOSITION  III. — If  the  numerator  of  a  fraction  be  di- 
vided by  any  number,  the  value  of  the  fraction  will  be 
diminished  as  many  times  as  there  are  units  in  the  di- 
visor. 

Examples. 


1.  Divide  f  f  by  2,  by  7. 

2.  Divide -Vg2- by  56. 

3.  Divide  £f  f-  by  25,  by  8. 


4.  Divide  f  jj  by  8,  16, 10. 

5.  Divide  $fc  by  2,  4,  8. 

6.  Divide  T4T2T  by  3,  21,  7. 


118.   Let  it  be  required  to  divide  T9T  by  3. 

ANALYSIS. — In  T9T  there  are   9  frac-  OPERATION. 

tional  units,  each  of  which  is  yL.  Now,  9  _±_  3  _  nxa  _  9  ^ 
if  we  multiply  the  denominator  by  3,  it 

becomes  33,  and  the  fractional  unit  becomes  3^,  which  is  only  £ 
of  TY,  because  33  is  3  times  as  great  as  11.  If  we  take  this 
fractional  unit  9  times,  the  result,  /5,  is  exactly  £  of  TTT :  hence, 
we  have 

PROPOSITION  IY. — If  the  denominator  of  a  fraction  be 
multiplied  by  any  number,  the  value  of  the  fraction  will 
be  diminished  as  many  times  as  there  are  units  in  the 
multiplier. 


117.  What  is  Proposition  HI.  ?— 118.  What  is  Proposition  IV  ? 


FUNDAMENTAL'  PROPOSITIONS.  £ 

Hence,  to  divide  a  fraction,  multiply  the  denominator. 
Examples. 


1.  Divide  -J  by  2. 

2.  Divide  \  by  7. 


3.  Divide  fj-   by  8. 

4.  Divide  £J  by  17. 


119.  Let  it  be   required   to   multiply  both   terms   of  the 
fraction  f  by  4. 

ANALYSIS. — In  £  the  fractional  unit  is  i,  and  it 
is  taken  3  times.    By  multiplying  the  denominator 
by  4,  the  fractional  unit  becomes  oV>  the  value  of     "sfxT  ==  T5 
which  is  £  times  as  great  as  j.     By  multiplying 
the  numerator  by  4,  we  increase  the  number  of  fractional  units 
taken,  4  times;   that  is,  we   increase   the  number  just  as  many 
times  as  we  decrease  the  value ;  hence,  the  value  of  the  fraction 
is  not  changed :  therefore,  we  have 

PROPOSITION  Y. — If  both  terms  of  a  fraction  be  multiplied 
by  the  same  number,  the  value  of  the  fraction  will  not  be 
changed. 

Examples. 

1.  Multiply  the  numerator  and   denominator  of  f  by  7  : 
this  gives,  -^  =  f  I- 

2.  Multiply  the  numerator  and  denominator  of  T7j  by  3, 
by  4,  by  5,  by  6,  by  9. 

3.  Multiply  each  term  of  |f  by  2,  by  3,  by  4,  by  5,  by  6. 

120.  Let  it  be  required  to  divide  the  numerator  and  de- 
nominator of  T65-  by  3. 

ANALYSIS.— In  Tfly  the  fractional  unit  is  Ty,  and      OPERATION 
is  taken  6  times.     By  dividing  the  denominator        e  -*-  8 
by  3,  the  fractional  unit  becomes  £,  the  value  of       15  +  3  —  i- 
which  is  3  times  as  great  as  yT.     By  dividing  the 
numerator  by  3,   we   diminish  the    number  of   fractional    units 
taken,  3  times ;    that  is,  we  diminish  the  number  just  as  mantf 
times  as  we  increase  the  value;  hence,  the  value  of  the  fraction 
is  not  changed:  therefore,  we  have 

119.  What  is  Proposition  V.  ? 


90  COMMON    FRACTIONS. 

PROPOSITION  VI.  —  If  both  terms  of  a  fraction  be  divided 
by  the  same  number,  the  value  of  the  fraction  will  not  be 
changed. 

Examples. 

1.   Divide  both  terms  of  the  fraction  T\  by  2  :   this  gives 


2.  Divide  both  terms  by  8  :  this  gives     ^|  =  J. 

3.  Divide  both  terms  of  the  fraction  T322g-  by  2,  by  4,  by 
8,  by  16. 

4.  Divide  both  terms  of  the  fraction  y6/^  by  2,  by  3,  by 
4,  by  5,  by  6,  by  10,  by  12. 

Reduction  of  Fractions. 

121.  REDUCTION  OF  FRACTIONS  is  the  operation  of  changing 
the  fractional  unit,  without  altering  the  value  of  the  fraction. 

122.  The  LOWEST  TERMS  of  a  fraction,   are   those   which 
are  prime  to  each  other. 

CASE    I. 

123.  To  reduce  a  whole  number  to  a  fraction  having  a 
given  denominator. 

1.   Reduce  6  to  a  fraction  whose  denominator  shall  be  4. 

ANALYSIS.  —  This  question  requires  us  to  re- 
duce   6,    to    fourths.     In   1    unit,    there   are  4          OPERATION. 
fourths;  in  6  units,  there  are  6  times  4  fourths,         6  X  4  =  24. 
or  24  fourths  :   therefore,   6  =  2T4.     Hence,  the  2^ 

following 

Rule 

Multiply  the  whole  number  and  denominator  together,  and 
write  the  product  over  the  required  denominator. 

120.  What  is  Proposition  VI.  ? 

121.  What  is  Reduction  of  Fractions? 

122.  What  are  the  lowest  terms  of  a  fraction  ? 

123.  What  is  Case  I.  ?    What  is  the  rule  ? 


REDUCTION.  91 

Examples. 

1.  Reduce  12  to  a  fraction  whose  denominator  shall  be  9. 

2.  Reduce  46  to  a  fraction  whose  denominator  shall  be  15. 


3.  Change  26  to  7ths. 

4.  Change  178  to  40ths. 

5.  Reduce  240  to  114ths. 


6.  Change  54  to  quarters. 

7.  Change  96  to  quarters. 

8.  Change  426  to  16ths. 


CASE    II. 

124.  To  reduce  a  mixed  number  to  its  equivalent  im- 
proper fraction. 

1.    Reduce  4J  to  its  equivalent  improper  fraction. 

ANALYSIS. — Since  in  any  number 

there  are  5  times  as  many  fifths  as  OPERATION. 

units  1,  in  4  there  will  be  5  times  4  X  5  =  20  fifths. 

4  fifths,  or  20  fifths,  to  which  add  Add      ...     4  fifths, 

4   fifths,    and   we    have    24    fifths.  iyes     _   ,4  =  ^  fifths. 
Hence,  the  following 

Rule. 

Multiply  the  whole  number  by  the  denominator  of  the 
fraction :  to  the  product  add  the  numerator,  and  place  the 
sum  over  the  given  denominator. 

Examples. 

1.  Reduce  47f  to  its  equivalent  fraction. 

2  In  17f  yards,  how  many  eighths  of  a  yard? 

3.  In  4259Q  rods,  how  many  twentieths  of  a  rod  ? 

4.  Reduce  625T4T  to  an  improper  fraction. 

5.  How  many  112ths  in  205T4r6s  ? 

6.  In  84 £J  days,  how  many  twenty-fourths  of  a  day? 

7.  In  15-J-Jf  years,  how  many  365ths  of  a  year? 

8.  Reduce  91 6^$-  to  an  improper  fraction. 

9.  Reduce  25Y9g-,  156||,  to  their  equivalent  fractions. 

124.  What  is  Case  II.  ?  How  do  you  reduce  a  mixed  number  to 
its  equivalent  improper  fraction? 


92  COMMON   FRACTIONS. 


CASE     III. 

125.    To   reduce    an  improper   fraction    to   its    equivalent 
whole  or  mixed  number. 

1.    In  -•/-,  how  many  entire  units  ? 

ANALYSIS. — Since  there  are  8  eighths  in  the  unit     OPERATION. 
1,  in  *-/  there  are  as  many  units,  as  8  is  contained         8)59 
times  in  59,  which  is  7|  times.     Hence,  the  fol-  ^3 

lowing 

Rule. 

Divide  the  numerator  by  the  denominator,  and  the  quo- 
tient will  be  the  whole  or  mixed  number. 

Examples. 

1.  Reduce  -^  and  -6g7-  to  their  equivalent  whole  or  mixed 
numbers. 

OPERATION.  OPERATION. 

4)84  9  )_67^ 

21  7J 

2.  Reduce  -9^-  to  a  whole  or  mixed  number. 

3.  In  Jy9-  yards  of  cloth,  how  many  yards  ? 

4.  In  -y-  bushels,  how  many  bushels  ? 

5.  If  I  give  \  of  an  apple  to  each  one  of  15  children, 
how  many  apples  do  I  give  ? 

6.  xveduce          ,         --,  — „        •,  }    to   their  whole 
or  mixed  numbers. 

7.  If  I  distribute  878  quarter-apples  among  a  number  of 
boys,  how  many  whole  apples  do  I  use  ? 

8.  Reduce    ^Y/J ,    4^g  ,    226748]463764,    to    their   whole   or 
mixed  numbers. 

9.  Reduce  JLil^LlLiij  L*J>lAO}  62"^735,  to  their  whole 
or  mixed  numbers. 

125.  What  is  Case  III.?    What  is  the  rule? 


REDUCTION. 


93 


CASE     IV. 
126.    To  reduce  a  fraction  to  its  lowest  terms. 


1ST  OPERATION. 

i )  ii  =  f  - 

2D   OPERATION. 
35  )  T7T°5    —   5" 


1.   Reduce  /T°y  to  its  lowest  terms. 

ANALYSIS.— By  inspection,  it  is  seen  that  5 
is  a  common  factor  of  the  numerator  and  de- 
nominator. Dividing  by  it,  we  have  £f .  We 
then  see  that  7  is  a  common  factor  of  14  and 
35:  dividing  by  it,  we  have  f ;  and  2  and  5 
'are  prime  to  each  other:  hence,  2  and  5  are 
the  lowest  terms. 

The  greatest  common  divisor  of  70  and  175 
is  35  (Art.  96)  ;   if  we  divide   both   terms  of 
the  fraction  by  it,  we  obtain  f .    The  value  of 
the  fraction  is  not  changed  in  either  operation,  since  the  numer- 
ator and  denominator  are  both  divided   by  the   same   number 
(Art.  120) :     Hence,  the  following 

Rule. 

Divide  the  numerator  and  denominator  by  each  of  their 
common  prime  factors,  in  succession : 

Or,  Divide  the  numerator  and  denominator  by  their 
greatest  common  divisor. 

Examples. 

Reduce  the  following  fractions  to  their  lowest  terms  : 


1. 

Reduce 

12 

IT' 

2. 

Reduce 

18 
24' 

3. 

Reduce 

27 
36' 

4. 

Reduce 

TfV 

5. 

Reduce 

84 

Te' 

6. 

Reduce 

TFT- 

7. 

Reduce 

283 
"2592 

8. 

Reduce 

85 
165' 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 

Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 

175 

sTT' 

104 

3Ti>' 

1049 
8T9T' 
275 
440* 
351 
795' 
172 
1113' 
63 
81' 
3  1  5 
405' 

12G.  What  is  Case  IV.?     How  do  you  reduce  a  fraction  to  its 
lowest  terms  ?    Is  the  value  of  the  fraction  altered  ?    Why  not  ? 


94  COMMON   FRACTIONS. 

CASE    V. 
127.   To  reduce  a  compound  fraction  to  a  simple  one. 

1.  What  is  the  value  of  f  of  f  ? 

ANALYSIS. — Three-fourths  of  -,,  is   3   times   1          OPERATION. 
fourth  of  4;    1  fourth  of  f,  is  ^  (Art.  118);   3          8x5  _  ,  5 
fourths   of  -f>   is   3   times  -^,   or  -|f :    therefore,          4  x  7 
I  of  4  =  |f.    Hence, 

Rule. 

Multiply  the  numerators  together  for  a  new  numerator, 
and  the  denominators  together  for  a  new  denominator. 

NOTE. — If  there   are  mixed   numbers,   reduce   them   to   their 
equivalent  improper  fractions. 

Examples. 

Reduce  the  following  fractions  to  simple  ones: 

4.  Reduce  2J  of  6J  of  7. 

5.  Reduce  5  of  j' of  i  of  6. 

6.  Reduce  6J  of  7J  of  6§f 


1.  Reduce  J  of  J  of  f. 

2.  Reduce  f  of  f  of  f . 

3.  Reduce  f  of  |  of  T93. 


Method  by  Canceling. 

128.  Since  the  numerators  are  factors  of  a  dividend,  and 
the  denominators  factors  of  a  divisor,  the  common  factors 
may  be  canceled.  When  they  are  all  canceled,  the  compound 
fraction  will  be  reduced  to  a  simple  fraction  in  its  lowest 
terms. 

Examples. 

1.   Reduce  f  of  j  of  T9j  to  a  simple  fraction. 

2         OPERATION. 

0        ft         0        2 
-  of  -  of  —  —  — 

$     0     n    5 

5 

127.  What  is  Case  V.  ?    How  do  you  reduce  a  compound  fraction 
to  a  simple  one  ? 

128.  How  are  compound  fractions  reduced  by  cancellation? 


REDUCTION.  95 

2..  Reduce  f  of  $  of  f  to  a  simple  fraction. 
Reduce  the  following : 


3.  f  of  f  of  a  of  ffr  of 
4-          of        of          of 


. 


5.    3f  of  f  of  3%  of  49. 


6.    ^  of  6^  of  f|. 
t.   J  of  |}  of  }  of  2f 
8.   if  of  T%  of  i|  of 


CASE     VI. 

129.  To  reduce  fractions  having  different  denominators 
to  equivalent  fractions  having  a  common  denominator. 

Fractions  have  a  common  denominator,  when  their  de- 
nominators are  alike. 

1.    Reduce  -J-,  £,  and  f,  to  a  common  denominator. 

ANALYSIS. — The  numerator  and  OPERATION 

denominator     of    each     fraction         1      _      _  

must  be  multiplied  by  the  same 

number,  else  the  value  will  be  7x2x5  =  70  2d  num. 
changed.  Take  the  product  of  4  X  2  X  3  =  24  3d  num. 
the  denominators  as  the  common  2x3x5  =  30  oom  den 
denominator.  Since  any  one  of 

these  denominators,  multiplied  by  the  product  of  the  other  two, 
will  give  this  common  denominator,  each  numerator  must  be  mul- 
tiplied by  the  same  product.  Multiplying  the  terms  of  ^  by  3 
and  5,  the  denominators  of  the  other  fractions,  we  have  ^| ; 
multiplying  the  terms  of  f  by  2  and  5,  the  denominators  of 
the  other  fractions,  we  have  §g  ;  and  multiplying  the  terms  of 
£  by  2  and  3,  the  denominators  of  the  other  fractions,  we  have 
f£:  Hence  the 

Rule. 

Multiply  the  numerator  of  each  fraction  by  all  the  denomi- 
nators except  its  own,  for  the  new  numerators,  and  all  the 
denominators  together  for  a  common  denominator. 

129.  What  is  Case  VI.  ?  What  is  a  common  denominator  ?  How 
do  you  reduce  fractions  of  different  denominators  to  fractions  hav- 
ing a  common  denominator?  When  the  numbers  are  small,  how 
may  the  work  be  performed? 


96  COMMON    FRACTIONS. 

NOTES. — 1.  Before  multiplying,  reduce  all  fractions  to  simple 
fractions. 

2.  When  the  numbers  are  small,  the  work  may  be  performed 
mentally :  Thus : 

i   *,    !,  =  $*,   H,   M- 

Examples. 

Reduce  the  following  fractions  to  common  denominators  : 


1.  Reduce  f ,  f,  and  T. 

2.  Reduce  f ,  T4r,  and  f . 

3.  Reduce  f ,  |,  and  f . 

4.  Reduce  2J,  and  \  of  j. 

5.  Reduce  5J,  -f  of  |,  and  4. 


6.  Reduce  3J  of  J,  and  |. 

7.  Reduce  J,  Y<r>  and  37- 

8.  Reduce  4,  fj,  and  -632-. 

9.  Reduce  7|,  f  J,  and  6J. 
10.  Reduce  4J,  8|,  and  2|. 


NOTE. — We  may  often  shorten  the  work  by  multiplying  the 
numerator  and  denominator  of  each  fraction  by  such  a  number 
as  will  make  the  denominators  the  same  in  all.  „ 

11.   Reduce  J  and  £,  to  a  common  denominator. 

OPEEATION. 

ANALYSIS. — Multiply  both  terms  of  the  first  by          j.  _  s 
8,  and  both  terms  of  the  second  by  2. 


12.  Reduce  J  and  J. 

13.  Reduce  |,  ^,  and  }. 

14.  Reduce  y,  -fa,  and  T4T. 


15.  Reduce  f,  3f,  and  f. 

16.  Reduce  6T5^,  9J,  and  5. 

17.  Reduce  7f,  |,  J,  and  J. 


CASE     VII. 

130.    To  reduce  fractions  to  their  least  common  denomi- 
nator. 

The   LEAST    COMMON    DENOMINATOR    is   the   number   which 
contains  -all  the   prime  factors  of  the  denominators. 

130.  What  is  the  least  common  denominator  of  several  fractions  ? 
How  do  you  reduce  fractions  to  their  least  common  denominator? 


REDUCTION.  97 


, 


1.   Reduce  £,  £,  and  f,  to  their  least  common  denominator. 

ANALYSIS. — The  least  common  multiple  of  the  denominators 
will  be  the  least  common  denominator,  and  in  the  example,  is 
12.  We  then  divide  12  by  each  denominator,  to  find  the  factor 
by  which  the  corresponding  numerator  must  be  multiplied,  that 
the  value  of  the  fraction  be  not  changed;  and  finally,  we  multi- 
ply each  numerator  by  its  proper  factor.  Therefore,  the  fractions 
5,  |,  and  f,  reduced  to  their  least  common  denominator,  nro 
&  it,  and  ft. 

OPERATION. 

3  )  3  .  6  .  4  (12  -^  3)  x  1  =    4,   1st  numerator. 

2  )  1  .  2  .  4          (12  -T-  6)  X  5  -  10,  2d 
1.1.2          (12  -f-  4)  X  3  =    9,  3d 
3x2x2  =  12,  least  com.  denominator. 

Hence,  the  following 

Rule. 

I.  Find  the  least  common  multiple  of  the  denominators 
(Art.  94),  which  will  be  the  least  common  denominator  of 
the  fractions. 

II.  Divide  the  least  common  denominator  by  the  denom- 
inator of  each  fraction,  separately ;  multiply  the  numerator 
by  the  corresponding  quotient,  and  place  each  product  over 
the  least  common  denominator. 

NOTE.— Before  beginning  the  operation,  reduce  every  fraction 
to  a  -simple  fraction,  and  to  its  lowest  terms. 

Examples. 

Reduce  the  following  fractions  to  their  least  common 
denominator : 


1.   Reduce 


,     ,       . 


2.  Reduce  14f,  6|,  5J. 

3.  Reduce  T»ffl  ^  f. 

4.  Reduce 


- 


5.    Reduce  jj.  3ft,  4. 


6.  Reduce  31    4ft,  8ft. 

7.  Reduce  J,  f,  j,  and.f. 

8.  Reduce  2^  of  £,  3}  of  2. 

9.  Reduce  f,  |,  J-,  and  TV 
10.  Reduce  },  },  f,  i  B- 


98  COMMON   FRACTIONS. 


ADDITION  OF  FRACTIONS. 

131.    ADDITION  OP  FRACTIONS    is  the   operation  of  finding 
the  sum  of  two  or  more  fractional  numbers. 

1  .  What  is  the  sum  of  ^,  f  ,  and  f  ? 

ANALYSIS.  —  The  fractional  unit  is  the 
same  in  each  fraction,  viz.:  |;  the  numer- 
ator  of  each  fraction  shows  how  many  r  3  4-  O  «=  9. 

such  units  are  taken:  hence,  the  sum  of  g  _ 

the   numerators,  written  over  the  common         •"****•  "?       42- 
denominator,  expresses  the  sum  of  the  frac- 
tions. 

2.  What  is  the  sum  of  l  and  f  ? 

ANALYSIS.—  In   the    first,   the    fractional  OPEKATION. 

unit  is  |,  in  the  second  it  is  ^.     These  units,         .     i  —  f 
not  being  of  the  same  kind,  cannot  be  ex-  j.  —  |. 

pressed   in   the   same   collection.     But  the 
£  =  f,   and   f  =  |,    in  each  of  which  the       f  +  |  =  J  =  1£. 
unit  is  £:  hence,  their  sum  is  |  =  1|. 


NOTE.  —  0?ifo/  wmte  o/  £A0  same   kind,   whether  fractional  or 
integral,  can  be  expressed  in  the  same  collection. 

From  the  above  analysis,  we  have  the  following 

Rule. 

I.  When  the  fractions  have  the  same  denominator,  add 
the  numerators,  and  place  their  sum  over  the  common  de- 
nominator. 

II.  When  they  have  not  the  same  denominator,  reduce 
them  to  a  common  denominator,  and  then  add  as  before. 


131.  What  is  Addition  of  Fractions?  When  the  fractional  unit 
is  the  same,  what  is  the  sum  of  the  fractions  ?  What  units  may 
be  expressed  in  the  same  collection  ?  What  is  the  rule  for  the 
addition  of  fractions  ? 


ADDITION. 


99 


Examples. 


1.  Add  i 

2.  Add  |, 

3.  Add  f, 


\,  |,  and  f 

f-,  and  f . 

>  |,  -§?->  and  Jg6-. 


4.  Add  T»T>  T8T,  T97,  and 

5.  Add  -I,  T3o,  and  ft- 


8.  Add  f,  f  J,  and  TV 

9.  Add  9,  f ,  &,  |,  and  f 

10.  Add  ^,  f,  f,  f  and  |. 

11.  Add  /Tl  f,  T%,  and  |. 

12.  Add  i   f,  and  f. 

13.  Add  y^,  f,  f,  and  f . 

14.  Add  T%,  f,  f ,  and  /Q. 


6.  Add  i,  f  ,  f  ,  and  T\. 

7.  Add  |,  f  ,  f,  and  TV 

15.  What  is  the  sum  of  191,  6f,  and  4|? 

OPERATION. 


Fractions. 
1  +  f  +  f  =  }-«£  = 


Whole  numbers. 
.    19  +  6  +  4  =  29; 

wim  =:  29 

132.    NOTE.  —  When  there  are  mixed  numbers,  a<Z<?  the  whole 
numbers  and  fractions  separately,  and  then  add  their  sums. 


16.  Add  3J,  7T%,  12|,  If 

1-7.  Add  16,  9J,  25J,  ft. 

18.  Add  i  of  |,  ^  of  9,  14TV 

19.  Add  2T8T,  6J,  and  12i|. 


20.  Add  900TV,  450J, 
21. 


22.  Add  17§  to  |  of  7f. 

23.  Add  f  7J,  8|. 


24.  What  is  the  sum  of  |  of  12§  of  7|,  and  f  of  25? 

25.  What  is  the  sum  of  ^  of  9f ,  and  ^4T  of  328§  ? 

133.    1.  What  is  the  sum  of  \  and  -J-? 

NOTE. — If  each  of  two  fractions 
has  1  for  a  numerator,  the  sum  of 
the  fractions  will  be  equal  to  the 
sum  of  their  denominators  divided 
by  their  product. 

2.  What  is  the  sum  of  %  and  J?     Of  £  and^? 

?     Of  A  and  A  ?     Of 


OPERATION. 

*+*=A+*«tt. 


3.  What  is  the  sum  of  \  and 


4.  What  is  the  sum  of  J 


and 


Of  £  and  J-  ?     Of 


and 


100  COMMON   FRACTIONS. 


SUBTRACTION. 

134.    SUBTRACTION  OF  FRACTIONS  is  the  operation  of  finding 
the  difference  between  two  fractions. 

1.  What  is  the  difference  between  f  and  f  ? 

ANALYSIS. — In  this  example,  the  fractional 
unit  is   1 :    there    are  5  such  units   in  the        &  OPERATION^ 
minuend  and  3  in  the  subtrahend :  their  dif-       T  ~~  t  —  "s  —  J* 
ference  is  2  eighths;  therefore,  2  is  written  Ans.  J. 

over  the  common  denominator  8. 


2.  From  -1/-  take 

3.  From  -|-  take  f . 


4.  From  Jf f  take 

5.  From  fff  take 


6.  What  is  the  difference  between  |-  and  J  ? 

OPERATION. 

ANALYSIS. — Reduce  both  to  the  same  s.  _  i^ 

fractional  unit,  TV ;  then,  there  are  10  such  ± _4 

units  in  the  minuend  and  4  in  the  subtra-       IQ        43    _  ]  J  _ 
hend :  hence,  the  difference  is  6  twelfths.       T^  ~  T2  —  T2  —  5- 

Ans.  \, 

From  the  above  analysis  we  have  the  following 
Rule. 

I.  When  the  fractions  have  the  same  denominator,  sub- 
tract the  less   numerator  from   the  greater,  and  place  the 
difference  over  the  common  denominator. 

II.  When  they  have  not  the  same  denominator,   reduce 
them    to    a    common    denominator,    and   then    subtract   as 
before. 

132.  When  there  are  mixed  numbers,  how  do  you  add? 

133.  When  two  fractions  have  1  for  a  numerator,  what  is  their 
sum  equal  to  ? 

134.  What  is  Subtraction  of  Fractions  ?    What  is  the  rule  ? 


SUBTRACTION.  101 


Examples. 


1.  From  f  take  |. 

2.  From  f  take  f . 

3.  From  ^  take  T5T. 


4.  From  1  take  T6oV 

5.  From  l  of  12,  take  j|  of  J. 

6.  Fm  f  of  1 J  of  7,  take  f  of  f. 


7.  From  f  of  f  of  J,  take  T3T  of  f  of  1. 

8.  From  f  of  |  of  6J,  take  f  of  £  of  f 
,  9.   From  T4r  of  ff  of  J,  take  T\  of  f  . 

10.  What  is  the  difference  between  41  and  2-J- 
OPERATION. 

or 


135.    Therefore  :   TFAen  ^/iere  are  mixed  numbers,  change 
to  improper  fractions,  and  subtract  as  in  Art.  134  : 
Or,  subtract  the  integral  and  fractional  numbers  separately. 

11.  From  84T75  take  16J.      |     12.  From  246f  take  164J. 
1.3.  From  7f  take  4J  :    f  =  ^6r  and  J  =  ?7T. 

NOTE.  —  Since  we  cannot  take  ^7T  from  ^T,  we       OPERATION. 
borrow  1,  or  |i,  from  the  minuend,  which,  added        ^2.  _  »?  6 


$  ~    ^ 


to    OT  =  |}  ;    then,   /T  from   |  j,   leaves  |  f     We 

must  now  carry  1  to  the  next  figure  of  the  sub- 

trahend,  and  proceed  as  in  subtraction  of  simple       Ans.   2--. 

numbers. 

14.  From  16|  take  5|.       j    16.  From  36|  take  27T8T. 

15.  From  26f  take  19J.         17.  From  400T5f  take  327|. 

18.  From  the  fraction  |,  take  the  fraction  j^ 
NOTE.  —  When  the  numerators  are  OPERATION 

1,  the   difference    of  the   two   frac-  JL  _   ^  =  JJL  __  B_  _  _3_ 
tions   is   equal  to  the   difference   of  8811_s8 

the    denominators   divided   by   their  1"~"  TT    :  :   TTxs     —  s's" 
product. 

19.  What  is  the  difference  between  1  and  J  ?    Between  J 
andTV?    }andTV?    ^  and  ^  ?    JT  and 


102  COMMON   FRACTIONS. 


MULTIPLICATION. 

136.  MULTIPLICATION  OF  FRACTIONS  is  the  operation  of 
taking  one  number  as  many  times  as  there  are  units  in 
another,  when  one  or  both  of  the  numbers  are  fractional. 

CASE    I. 
137.    To  multiply  a  fraction  by  a  whole  number. 

1.  If  one  yard  of  cloth  cost  f  of  a  dollar,  what  will  4 
yards  cost  ? 

ANALYSIS. — Four  yards  will  cost 

-         ~.  OI rVH-KATION. 

4  times  as  much  as  1  yard.     Since 

1  yard  costs  5  eighths  of  a  dollar,  4      ¥  x  ^  —  ~r~  —  ~T~ ~  ^  J. 

yards  will  cost  4  times  5  eighths  of 

a  dollar,  which  is  20  eighths:  therefore,  if  1  yard  cost  f  of  a 

dollar,  4  yards  will  cost  ~  =  2£  dollars. 

2d.   If  we  divide  the  denominator  by  OPERATION 

4,   the  fraction  will  be  multiplied  by  4  '    5 

(Prop.  II.) :  performing  the  operation,  we  f  X  4  =  g-^-j  =  f . 
obtain  f ,  which  =  2^ :     Hence, 

To  multiply  a  fraction  by  a  whole  number, 
Multiply  the  numerator,  or  divide  the  denominator. 
Examples. 


1. 

2. 
3. 

Multiply 
Multiply 
Multiply 

T¥T  by 
ttby  > 

•W-  by 

12. 
9. 

4. 
5. 
6. 

Multiply 
Multiply 
Multiply 

Ji¥ 

m 

]  75 
273 

by 
by 

5. 
49. 
26. 

135.  When  there  are  mixed  numbers,  how  do  you  subtract  ?    Ex- 
plain the  case  when  the  fractional  part  of  the  subtrahend  is  the 
greater. 

136.  What  is  Multiplication  of  Fractions  ? 

137.  What  is  Case  I.?    What  is  the  rule? 


MULTIPLICATION.  103 

T.  If  1  dollar  will  buy  £  of  a  cord  of  wood,  how  much 
will  15  dollars  buy? 

8.  At  |  of  a  dollar  a  pound,  what  will  12  pounds  of  tea 
cost  ? 

9.  If  a  horse  eats  f  of  a  bushel  of  oats  in  a  day,  how 
much  will  18  horses  eat  ? 

10.  What  will  64  pounds  of  cheese  cost,  at  -fa  of  a  dol- 
lar a  pound  ? 

11.  At  2 1  cents  a  pound,  what  will  8  pounds  of  chalk 
cost? 

NOTE. — When  the  multiplicand  is  a  mixed  number,  multiply 
the  fraction  and  integer  separately,  and  add  the  results ;  or,  re- 
duce the  mixed  number  to  an  improper  fraction,  and  multiply. 

12.  If  a  man  receires  3T^  dollars  a  day,  how  much  will 
he  receive  in  15  days  ? 

13.  If  a  family  consumes  5|  barrels  of  flour  in  1  year,  how 
much  would  they  consume  in  9  years  ? 

CASE    II. 
138.    To  multiply  a  whole  number  by  a  fraction. 

1.   At  15  dollars  a  ton,  what  will  f  of  a  ton  of  hay  cost  ? 

ANALYSIS. — 1st.  Four-fifths  of  a  ton  will  cost 
4  times  as  much  as  1  fifth  of  a  ton;  if  1  ton  OPERATION. 

cost  15  dollars,  1  fifth  will  cost  \  of  15  dol-        */-  X  4  =  12. 
lars,  or  3  dollars,  and  £  will  cost  4  times  3 
dollars,  which  are  12  dollars. 

Or,  2d.  4  fifths  of  a  ton  will  cost  1  fifth  of 
4  times  the  cost  of  1  ton ;  4  times  15  is  60,  ^ 

and  1  fifth  of  60  is  12:    Hence,  **  x  4  —  12. 

0 
Rule. 

Divide  the  whole  number  by  the  denominator  of  the 
fraction,  and  multiply  the  quotient  by  the  numerator: 

Or,  Multiply  the  whole  number  by  the  numerator  of  the 
fraction,  and  divide^,  the  product  by  the  denominator. 

NOTE. — Cancel,  when  possible. 


104:  COMMON    FRACTIONS. 

Examples. 


1.  Multiply  24  by  J. 

2.  Multiply  42  by  £J 


3.  Multiply  105  by  f 

4.  Multiply  64  by  -Jf . 


5.  What  is  the  cost  of  f  of  a  yard  of  cloth,  at  8  dollars 
a  yard  ? 

6.  If  an  acre  of  land  is  valued  at  75  dollars,  what  is  ^ 
of  it  worth  ? 

7.  If  a   house   is   worth   320   dollars,  what   is   T9¥   of  it 
worth  ? 

8.  If  a  man  travels  46  miles  in  a  day,  how  far  does  he 
travel  in  -f  of  a  day? 

9.  At  18  dollars  a  ton,  what  is  the  cost  of  T9^  of  a  ton 
of  hay  ? 

10.  If  a  man  earn  480  dollars  in  a  year,  how  much  does 
he  earn  in  ^-J  of  a  year? 

CASE    III. 
139.    To  multiply  one  fraction  by  another. 

1.  If  a  bushel  of  corn  costs  f  of  a  dollar,  what  will  J 
of  a  bushel  cost? 

OPERATION. 

ANALYSIS. — 5   sixths   of  a  bushel  will         |.  x  s.  _  i  s.  _  5. . 
cost  £  times  as  much  as  1  bushel,  or  5 
times   1    sixth   as   much :   -g-  of  f   is  •£% 
(Art.  127),   and  5   times  &,  is  ±%  =  | :  #  x  <>  _  5. 

Hence,  4       0 

Rule. 

Multiply  the  numerators  together  for  a  new  numerator, 
and  the  denominators  together. for  a  new  denominator. 

NOTES. — 1.  When  the  multiplier  is  less  than  1,  we  do  not  take 
the  whole  of  the  multiplicand,  but  only  such  a  part  of  it  as  the 
multiplier  is  of  1. 

2.  When  the  multiplier  is  a  proper  fraction,  multiplication  does 

138.  What  is  Case  II.  ?    What  is  the  rule  ? 

139.  What  is  Case  III.?    What  is  the  rule? 


MULTIPLICATION.  105 

not  imply  increase,  as  in  the  multiplication  of  whole  numbers. 
The  product  is  the  same  part  of  the  multiplicand  which  the 
multiplier  is  of  1. 

3.  If  the  multiplicand  or  multiplier,  or  both,  be  whole  or  mixed, 
the  whole  number  may  be  reduced  to  a  fractional  form,  and  the 
mixed  numbers  reduced  to  improper  fractions;  and  then  the  last 
rule  will  apply  to  all  examples. 

Examples. 

1.    Multiply  I  by  f  |      2.   Multiply  &  by  £f. 

3.  Find  the  product  of  f,  f,  and  T7?. 

4.  Find  the  product  of  f ,  T9T,  and  §-J-. 

5.  If  silk  is  worth  T9^  of  a  dollar  a  yard,  what  is  -J  of  a 
yard  worth? 

6.  If  I  own  |  of  a  farm,  and  sell   f   of  my  share,  what 
part  of  the  whole  farm  do  I  sell  ? 

7.  At  f  of  a  dollar  a  pound,  what  will  ^  of  a  pound  of 
tea  cost  ? 

8.  If  a  knife  costs  f  of  a  dollar,  and  a  slate  f  as  much, 
what  does  the  slate  cost  ? 

OPERATION. 

9.  Multiply  5  J  by  J  of  |,  5  J  =  ^ ;     1  of  f  =  /7. 

7       2 

NOTE.  —  Before    multiplying,  £%        $ 

reduce    both    fractions    to    the  ~j  X  ^2  =  J' 

form  of  simple  fractions.  *«* 

9 
General  Examples. 


1.  Mult.  J  of  |  of  -J,  by  T9T. 

2.  Mult.  T%  by  |  of  1J. 

3.  Mult,  j-  of  3,  by  £  of  15J. 


4.  Mult.  5  of  f  of  f ,  by  4J. 

5.  Malt.  14  of  f  of  9,  by  6f 

6.  Mult,  f  of  6  of  f ,  by  f  of  4. 


139.  How  do  you  multiply  one  fraction  by  another?  When  the 
multiplier  is  less  than  1,  what  part  of  the  multiplicand  is  taken  ? 
If  the  fraction  is  proper,  does  multiplication  imply  increase  ?  What 
part  is  the  produ'ct  of  the  multiplicand? 

5* 


106  COMMON   FRACTIONS. 

140.  "When  the  multiplicand  is  a  whole,  and  the  multi- 
plier a  mixed  number. 

7.  What  is  the  product  of  48  by  8£? 

OPERATION. 

NOTE.— First  multiply  48  by  -£,  which  gives  43  x  i.  —  g 
8;  then  by  8,  which  gives  384,  and  the  sum  48  x  8  =  384 
392  is  the  product:  Hence, 

392 

Rule. 

Multiply  first  by  the  fraction,  and  then  by  the  whole 
number,  and  add  the  products. 


8.  Multiply  67  by  9J. 

9.  Multiply  9  by  12f. 


10.  Multiply  108  by  12j. 

11.  Multiply  5f  by  3i. 


12.  What  is  the  product  of  6J,  2J,  and  £  of  12  ? 

13.  What  will  24  yards  of  cloth  cost,  at  3|  dollars  a  yard? 

14.  What  will  6|  bushels  of  wheat  cost,  at  3j  doUars  a 
bushel  ? 

15.  A  horse  eats  T3T  of  |    of  12  tons  of  hay  in  three 
months :  how  much  did  he  consume  ? 

16.  If  f  of  f  of  a  dollar  buy  a  bushel  of  corn,  what  will 
rV  of  T6r  of  a  bushel  cost  ? 

It.  What  is  the  cost  of  5§  gallons  of  molasses,  at  96J 
cents  a  gallon  ? 

18.  What  will  7f£  dozen  candles  cost,  at.T\  of  a  dollar 
per  dozen  ? 

19.  What  must  be  paid  for  175  barrels  of  flour,  at  7f 
dollars  a  barrel  ? 

20.  If  |  of  -f-  of  2  yards  of  cloth  can  be  bought  for  one 
dollar,  how  much  can  be  bought  for  |-  of  13 J  dollars  ? 

21.  What  is  the  cost  of  15f  cords  of  wood,  at  3f  dollars 
a  cord  ? 


140.  How  may  you  multiply,  when  the  multiplicand  is  a  whole, 
and  the  multiplier  a  mixed  number? 


DIVISION.  107 


DIVISION. 

141.  DIVISION  OF  FRACTIONS  is  the  operation  of  finding 
how  many  times  one  number  is  contained  in  another,  when 
one  or  both,  are  fractional. 

1.  What  is  the  quotient  of  5  divided  by  -J-? 

ANALYSIS. — One-sixth  is  contained  in 

1,  6  times,  because  there  are  6  sixths  OPERATION. 

in    1 :    one-sixth   is    contained    in    5,       5  -r-  •!•  =  5  X  6  =  30. 
5  times  as  many  times  as  in  1 :  hence, 
y  is  contained  in  5,  30  times. 

2.  How  many  times  is  J  contained  in  8  ? 

3.  How  many  times  is  -J-  contained  in  6? 

4.  How  many  times  is  J  contained  in  9  ? 

CASE    I. 
142.    To  divide  a  fraction  by  a  whole  number. 

1.  If  4  bushels  of  apples  cost  £  of  a  dollar,  what  will  1 
bushel  cost  ? 

ANALYSIS. — Since  4  bushels  cost  £  of  a  OPERATION 

dollar,  1  bushel  will  cost  |  of  f  of  a  dollar.  8    .    .  , 

Dividing  the  numerator  of  the  fraction  by  ""  »" 

4,  we  have  f  (Art.  117). 

Multiplying  the    denominator  by  4,   will       s 
produce  the  same  result  (Art.  118):   Hence,       * 

Rule. 

Divide  the  numerator,  or  multiply  the  denominator,  by 
the  divisor. 

141.  What  is  Division  of  Fractions  ?  What  is  the  quotient  of  8 
divided  by  1? 

143.  What  is  Case  I.?    What  is  the  rule? 


108  COMMON   FRACTIONS. 


Examples. 


1.  Divide  }{.  by  6. 

2.  Divide  £f  by  9. 

3.  Divide  -4T°9A  by  15. 

4.  Divide  jf  f  by  75. 


5.  Divide  jf  by  6. 

6.  Divide  |f  by  12. 

7.  Divide  £f  by  20. 

8.  Divide  -iff  by  27. 


9.  If  6  horses  eat  T9^  of  a  ton  of  hay  in  1  month,  how 
much  will  one  horse  eat  ? 

10.  If  9  yards  of  ribbon  cost  f  of  a  dollar,  what  will  1 
yard  cost  ? 

11.  If  1  yard  of  cloth  cost  4  dollars,  how  much  can  be 
bought  for  f  of  a  dollar  ? 

12.  If  5  pounds  of  coffee  cost  -}f  of  a  dollar,  what  will 
1  pound  cost  ? 

13.  At  $6  a  barrel,  what  part  of  a  barrel  of  flour  can 
be  bought  for  f  of  a  dollar? 

14.  If  10  bushels  of  barley  cost  3|  dollars,  what  will  1 
bushel  cost  ? 

NOTE. — Reduce  the  mixed  number  to  an  improper  fraction, 
and  divide  as  in  the  case  of  a  simple  fraction. 

15.  If  21  pounds  of  raisins  cost  4f  dollars,  what  will  1 
pound  cost  ? 

16.  If  12  men  consume  6f  pounds  of  meat  in  a  day,  how 
much  does  1  man  consume  ? 

CASE    II. 
143.    To  divide  a  whole  number  by  a  fraction. 

1.    At  |  of  a  dollar  apiece,  how  many  hats  can  be  bought 
for  6  dollars? 

ANALYSIS. — As  many  as  f  of  a  OPERATION. 

dollar   is    contained   times    in    6      c_L.4._c_iv4 
dollars,     f  =  £  x  4.     To  divide  6 

by  |,  is  to  divide  it  by  &  and      =  6  x  I  =  4   =  H  na"s- 
then  the  quotient  by  4  (Art.  76). 

6-r|=:6x5  =  30;   then,  dividing  by  4,  we  have  V  =  7£  for 
the  answer.     Hence, 


DIVISION.  109 

Rule. 

Invert  the  terms  of  the  divisor,  and  multiply  the  whole 
number  by  the  new  fraction. 


Examples. 


1.  Divide  H  by  J. 

2.  Divide  212  by  } 


3.  Divide  63  by  £f. 

4.  Divide  420  by  T9T. 


5.  At  -JJ  of  a  dollar  a  yard,  how   many  yards   of  cloth 
can  be  bought  for  9  dollars  ? 

6.  If  a  man  travel  J  of  a  mile  in  1  hour,  how  long  will 
it  take  him  to  travel  10  miles  ? 

7.  If  |  of  a  ton  of  hay  is  worth  9  dollars,  what  is  a  ton 
worth  ? 

CASE    III. 
144.     To  divide  one  fraction  by  another. 

1.  At  f  of  a  dollar  a  gallon,  how  much  molasses  can  be 
bought  for  -§-  of  a  dollar? 

ANALYSIS. — As  many  times  as  OPERATION. 

f  of  a  dollar  is  contained  times     J._i.j  =  j._i_jx2 
in    |    of   a    dollar :    f  =  £  x  2  ;      _  _3_s  _^_  2  —  35.  _  2  s    ffan 
hence,  to  divide  by  f ,  is  to  divide 

by  £  and  2.     £  -f-  \  =  V  >  tnen  dividing  by  2,  we  have,  ^  =  2T^ 
gallons.    Hence, 

Rule. 

1.  Invert  the  terms  of  the  divisor : 

II.  Then  multiply  the  numerators  together  for  the  nume- 
rator of  the  quotient,  and  the  denominators  together  for  the 
denominator  of  the  quotient. 

MOTES. — 1.   Cancel  all  common  factors. 

2.  If  the  dividend  and  divisor  have  a  common  denominator, 
they  will  cancel,  and  the  answer  will  be  the  quotient  of  their 
numerators. 

3.  When  the  dividend  or  divisor  contains  a  whole  or  mixed 
number,  or  compound  fractions,  reduce  to  the  form  of  simple 
fractions,  before  dividing. 


110  COMMON   FRACTIONS. 


Examples. 


1.  Divide  TV  by  Jf. 

2.  Divide  T4T  by  £f. 

3.  Divide  3J  by  £[. 


4.  Divide  £  of  J  by  ^  of  1J. 

5.  Divide  f  of  21  by  |  of  3|. 

6.  Divide  6J  by  2J. 


7.  At  -J-  of  a  dollar  a  pound,  how  much  butter  can  be 
bought  for  }  J  of  a  dollar  ? 

8.  If  1  man  consume  1}  pounds  of  meat  in  a  day,  how 
many  men  would  8f  pounds  supply? 

9.  If  6  pounds  of  tea  cost  4J  dollars,  what  does  it  cost 
a  pound  ? 

10.  At   £   of  a   dollar   a  basket,   how  many  baskets   of 
peaches  can  be  bought  for  11 }  dollars? 

11.  If  f  of  a  ton  of  coal  cost  6|  dollars,  what  will  1  ton 
cost,  at  the  same  rate  ? 

12.  How  much  cheese  can  be  bought  for  ^f  of  a  dollar, 
at  J  of  a  dollar  a  pound  ? 

13.  A  man  divided  2f  dollars  among  his  children,  giving 
them  -£§  of  a  dollar  apiece ;  how  many  children  had  he  ? 

14.  How  many  times  will  jj  of  a  gallon  of  beer  fill  a 
vessel  holding  J  of  f  of  a  gallon  ? 

15.  How  many  times  is»-J-  of  ^  of  27  contained  in  J  of  J 
of  42| ? 

16.  If  5^  bushels  of  potatoes  cost  2|  dollars,  how  much 
do  they  cost  a  bushel  ? 

17.  If  John  can  walk  21  miles  in  |  of  a  day,  how  far  can 
he  walk  in  1  day? 

18.  If  a  turkey  cost  If  dollars,  how  many  can  be  bought 
for  12|  dollars  ? 

19.  At  f  of  |  of  a  dollar  a  yard,  how  many  yards  of 
ribbon  can  be  bought  for  fj  of  a  dollar  ? 

143.  What  is  Case  II.  ?     What  is  the  rule  ? 

144.  What  is  Case  III.?  What  is  thte  rule? 


COMPLEX   FRACTION'S. 


: 


111 


Complex  Fractions. 

145.    Complex  fractions  are  reduced  to  their  simplest  form, 
by  the  operations  of  Reduction  and  Division. 

2^ 
1.   Reduce    ~    to  its  simplest  form. 

OPERATION. 


3 

Hence,  for  the  reduction  of  a  complex  fraction  to  its  sim- 
plest form,  we  have  the  following 

Rule. 

Reduce  each  term  to  a  simple  fraction,  and  then  perform 
the  division. 

Examples. 

Reduce  the  following  complex  fractions  to  their  simplest 
form: 


1.   Reduce 

4 
f 

6. 

Reduce 

25 

8f 

2.  Reduce 

7 

f 

T. 

Reduce 

!f& 

|  of  15 

• 

3.  Reduce 

yi 

8. 

Reduce 

214f 
25H' 

4.   Reduce 

f  of 

f. 

9. 

Reduce 

ff- 

5.   Reduce 

f  of 

i 

10. 

Reduce 

f  off 

of  5| 

iof 

a* 

Hof 

48    ' 

145.  What  is  a  complex  fraction?     How  are  complex  fractions 
reduced  to  their  simplest  forms?    What  is  the  rule  for  Reduction? 


112  COMMON   FRACTIONS. 


Miscellaneous  Examples. 

1.  A  man,  having  9f  dollars,  paid  3|-  dollars  for  boots, 
and  4f  dollars  for  a  hat :  how  much  had  he  left  ? 

2.  A   retailer   gave   his   customer    IJ   dollars   in   change, 
which,  he   afterwards  found,  was  f   of  a  dollar  too  much: 
what  was  the  exact  amount  of  change  due  ? 

3.  A  young  clerk,  having  charged  f  of  a  dollar  too  much 
for  some   cloth,   gave  in  change,  If-  dollars :   what  was  the 
exact  amount  that  he  ought  to  have  given  ? 

4.  A  bank  of  issue  failed,  and  was   able   to  redeem  its 
notes  by  paying  f  of  a  dollar  on  a  dollar :  how  much  would 
he  who  has  a  10  dollar  bill,  receive  from  the  bank  ? 

5.  The  sum  of  two  numbers  is  12T7g-;  one  of  the  numbers 
is  7-f- :  what  is  the  other  ? 

6.  James,  Joseph,  and  Daniel  owned  three   farms,  whose 
total   area  was  475T9Q    acres.     Daniel  had   15f   acres   more 
than  Joseph,  and  Joseph  24|-  acres  more  than  James :  how 
many  acres  had  each  in  his  farm  ? 

7.  A  housekeeper  bought  6  mahogany  chairs,  at  3J  dol- 
lars each,  and  gave  for  them,  2  ten-dollar  and  1  five-dollar 
bill :  what  change  ought  she  to  receive  ? 

8.  A  mechanic  that  was  fond  of  reading,  wished  to  buy 
Macaulay's    History,   worth    6-£   dollars,    Irving's    Columbus, 
worth  4}  dollars,  and  Prescott's  Philip  II.,  worth  5f  dollars ; 
his  daily  wages  were  If  dollars  a  day  :  how  many  days'  wages 
would  pay  for  the  books  ? 

9.  If  12  barrels  of  flour  were  given  for  a  piece  of  cloth, 
measuring  31J  yards,  and  valued  at  2f  dollars  a  yard,  what 
would  be  the  value  of  one  barrel  ? 

10.  A  grocer  having  J  of  a  barrel  of  sugar,  sold  f  of  it 
for  4|  dollars:   what  was   the  value  of  the  barrel,  at  the 
same  rate  ? 

11.  The  product  of  f   of  2-},  by  f   of  f  of  9,  is   how 
much  greater  than  the  quotient  of  7£  divided  by  J  of  frj-  ? 


MISCELLANEOUS   EXAMPLES.  113 

12.  The  cost  of  a  barrel  of  flour  is  6}  dollars,  and  it  will 
buy  2  barrels  of  apples,  each  of  which  is  worth  1-J  barrels 
of  potatoes  :  how  many  pounds  of  butter,  at  |-  of  a  dollar  for 
3  pounds,  would  pay  for  a  barrel  of  potatoes  ? 

13.  The  product  of  3  numbers  is  -|  :  two  of  the  numbers 
are  2|  and  J  :    what  is  the  third  ? 

14.  A  father  and  son,  working  an  equal  number  of  days, 
earned  54|  dollars  :   the  father  received  If  dollars,  and  the 
son  ^  of  a  dollar,  a  day  :   how  many  days  did  they  work  ? 

15.  A  regiment  lost  in  battle  250  men,  which  was  J  of 
the   regiment  :    what   was    the    number    of   men    before    the 
battle  ? 

16.  A  merchant  owning  -fa  of  a  vessel,  sold  J  of  his  share 
for  1610  dollars  :    what  was  the  value  of  the  ship,  at  that 
rate? 

17.  How  many  lemons,  at  -fs  of  a  dollar  a  dozen,  will  pay 
for  81  oranges  at  2|  cents  each? 

18.  A  lad,  multiplying  by  f  instead  of  T\,  obtained  f  for 
a  result  :  what  result  ought  he  to  have  obtained  ? 

19.  Reduce  9         7  X  -          to  a  simple  fraction. 

^9  2   Ot   6" 

20.  If  |  of  a  yard  of  cloth  cost  J  of  a  dollar,  what  will  be 
the  cost  of  2f  yards  ? 

21.  If  23J  dollars  are  required  to  pay  18  men  for  1  day's 
wages,  how  much  would  be  required  to  pay  33  men  for  15  j 
days'  labor? 

22.  If  A.  can  mow  an  acre  of  ground  in  3  days,  and  B. 
in  2  days,  how  long  would  it  take  them  both  to  mow  it? 

23.  If  A.  and   B.  can  do   a  piece   of  work  in   10  days, 
and  A.  alone  can  do   it  in  J6  days,  in  what  time  can  B. 
do  it? 


24.  Multiply  by 

25.  In  a  piece  of  cloth  there  were  36J  yards.    The  piece 
cost  65J  dollars.     For  what  must  the  cloth  be  sold  at  per 
yard,  that  there  may  be  a  gain  of  18225  dollars  ? 


114  DECIMAL    P^RACTIONS. 


DECIMAL   FRACTIONS. 

1 46.  There  are  two  kinds  of  Fractions :   Common  Frac- 
tions, and  Decimal  Fractions. 

147.  A  COMMON  FRACTION,  is  one  in  which  the  unit  is  di- 
vided into  any  number  of  equal  parts. 

148.  A  DECIMAL  FRACTION,  is  one   in  which   the   unit   is 
divided   into    10    equal   parts,   then   each   of  these   parts   is 
again  divided  into  10  equal  parts,  and  so  on,  using  10  con- 
stantly as  a  divisor. 

When  the  unit  is  divided  into  10  equal  parts,  there  are 
10  such  parts  of  the  unit,  and  each  part  is  called,  one-tenth. 

If  each  tenth  be  divided  into  10  equal  parts,  there  will  be 
100  equal  parts  in  the  unit,  and  each  part  will  be  j1^  of  ^ 

—  loo- 

If  each  hundredth  be  divided  into  10  equal  parts,  there 
will  be  1000  equal  parts  in  the  unit,  and  each  part  will  be 
ro  °f  ii^  —  ToW  5  an(^  smaller  parts  may  be  obtained,  by 
still  dividing  by  10. 

Notation  and  Numeration. 


149.   A  period  (.),  called  the  decimal  point,  written  before 
i  figure,  denotes  that  its  unit  is  1  tenth  : 

Thus,       .1        is  read, 

1  tenth   =  ^ 

A 

4  tenths  =  T%. 

.7 

7  tenths  =  TV 

&c. 

&c. 

146.  How  many  kinds  of  Fractions  are  there  ?    What  are  they  ? 

147.  What  is  a  Common  Fraction  ? 

148.  What  is  a  Decimal  Fraction?    When  the  unit  is  divided 
into  10  equal  parts,  what  is  each  part  called  ?    What  is  each  part 
called,  when  it  is  divided  into  100  equal  parts? 


NOTATION   AND    NUMERATION.  115 

The  second  place  from  the  decimal  point,  is  the  place  of 
hundredths : 

Thus,       .01  is  read,  1  hundredth   =  T^. 

.04              "  4  hundredths  =  T^. 

.07              "  7  hundredths  =  Tfo. 
&c.,                                         &c. 

The  third  place  is  the  place  of  thousandths : 

Thus,    .001  is  read,         1  thousandth   =  T^T. 

.    .004  "              4  thousandths  =  ^v. 

.007  "              7  thousandths  =  T^. 

The  fourth  place  is  the  place  of  ten-thousandths ;  the 
fifth,  of  hundred-thousandths  ;  the  sixth,  of  millionths,  &c. 

Thus,  4,  written  in  the  different  places,  is  read, 

Four  tenths, 4 

Four  huudredths,        .       .       •.       .       .       .     .04 

Four  thousandths, 004 

Four  ten-thousandths, 0004 

Four  hundred-thousandths, 00004 

Four  millionths, 000004 

Four  ten-millionths, 0000004 

150.  We   numerate  from  the   decimal  point  to  the  right, 
and  read  in  the  lowest  fractional  unit  of  the  decimal.    Thus, 
we  numerate,  tenths,  hundredths,  &c. ;   and  read,    4  tenths, 
4  hundredths,  &c. 

151.  From   the   nature  -of  decimals,   and   the  manner   of 
writing  them,  we  see, 

1st.  That  the  denominator  belonging  to  any  decimal 
fraction,  is  1,  with  as  many  ciphers  annexed  as  there  are 
places  of  figures  in  the  decimal. 

149.  What  is  the  decimal  point  ?    Where  is  it  written  ?    What 
does  it  denote?    What  is  the  first  place  to  the  right  called?    The 
second?    The  third? 

150.  How  do  you  numerate  decimals?    How  do  you  read  them? 


116  DECIMAL  FRACTIONS. 

2d.  That  the  unit  of  any  place,  is  ten  times  as  great  as 
the  unit  of  the  next  place  to  the  right — the  same  as  in 
whole  numbers:  hence,  whole  numbers  and  decimals  may  be 
written  together,  by  placing  the  decimal  point  between  them, 
as  in  the  following 

Numeration  Table. 


2  $  .15 

02  O       £  02       3 

.2  ^       *  •     ^       |  J4 

*  i  j  • !  i I  rfjillifl 

OQ.S'g       w       pj'jS       02       H     73     ^       9     *f    TO     .2      >? 

83630642-0478976 


Whole  numbers.  Decimals. 

152.  A  MIXED   NUMBER,  is   composed  partly  of  a  whole 
number,  and  partly  of  a  decimal :    Thus,  67.0478  is  a  mixed 
number. 

Rule  for  Notation  and  Numeration. 

153.  I.  Write  the  decimal  as  if  it  were  a  whole  number, 
and  then  prefix  as  many  ciphers  as  may  be  necessary  to 
give  the  true  name  to  the  last  significant  figure  ;  and  then 
prefix  the  decimal  point. 

II.  Head  the  decimal  in  terms  of  its  lowest  unit,  the  same 
as  if  it  were  a  whole  number. 


151.  What  is  the  first  principle  which  follows  from  the  nature 
of  decimals,  and  the  manner  of  writing  them  ?    What  is  the  second 
principle  ?    What  follows  from  this  principle  ? 

152.  What  is  a  mixed  number? 

153.  Give  the  rule  for  Notation.     Give  the  rule  for  Numeration. 


NOTATION    AND   NUMERATION. 


117 


Examples. 

Write  the  following  numbers  decimally : 


(10 

3 
100 

(6.) 


(2.) 

16 

1000 


(3.) 

IT 

10000 

(8.) 


32 
100 

(9.) 


(10-.) 

10W 


Write  the  denominators  belonging  to  the   following  deci- 
mals : 

(11.)  (12.)  (13.)                  (14.) 

.0479  .4756  .0001               .674124 


(15.) 
.2700 


(16.) 
.47043 


(17.) 
.270496 


Numerate  and  read  the  following : 

(19.)  (20.)  (21.) 

67.0472  2.00498  6.010406 


(18.) 
.000047 

(22.) 
1.890470 


Write  the  following  numbers  in  figures,  and  then  numerate 

them :  also,  write  the  denominator  of  each  : 

23.  Forty-one,  and  three-tenths. 

24.  Sixteen,  and  three  millionths. 

25.  Five,  and  nine  hundredths. 

26.  Sixty-five,  and  fifteen  thousandths. 

27.  Eighty,  and  three  millionths. 

28.  Two,  and  three  hundred  millionths. 

29.  Four  hundred,  and  ninety-two  thousandths. 

30.  Three  thousand,  and  twenty-one  ten  thousandths. 

31.  Forty-seven,  and  twenty-one  hundred  thousandths. 

32.  Fifteen  hundred,  and  three  millionths. 

33.  Thirty-nine,  and  six  hundred  and  forty  thousandths. 

34.  Three  thousand,  eight  hundred  and  forty  millionths. 

35.  Six  hundred  and  fifty  thousandths. 


118  DECIMAL   FRACTIONS. 

154.  Annexing  Ciphers. 

ANNEXING  a  cipher  is  placing  it  on  the  right  of  a 
number. 

If  a  cipher  is  annexed  to  a  decimal,  it  makes  one  more 
decimal  place;  and  therefore,  a  cipher  must  also  be  annexed 
to  the  denominator  (Art.  151). 

The  numerator  and  denominator  will  therefore  have  been 
multiplied  by  the  same  number,  and  consequently  the  value 
of  the  fraction  will  not  be  changed  (Art.  119) :  Hence, 

Annexing  ciphers  to  a  decimal  does  not  alter  its  value. 
We  may  take  as  an  example,     .3  —  T3^. 

If  we  annex  a  cipher  to  .3,  we  must,  at  the  same  time, 
annex  one  to  the  numerator  and  denominator  of  T^ ;  thus, 

.3  —  T%  =  T3o°^  =  .30,  by  annexing  one  cipher. 

•3  =  T3o  =  T3o°o  =  iTO)  =  -300'  b?  annexing  two  ciphers. 

In  like  manner,  any  decimal  may  be  changed  from  a  higher 
to  a  lower  fractional  unit,  without  altering  its  value. 

Also,  if  a  decimal  point  be  placed  on  the  right  of  an  in- 
tegral number,  and  ciphers  be  then  annexed,  the  value  will 
not  be  changed  :  thus,  5  =  5.0  =  5.00  =  5.000,  &c. 

155.  Prefixing  Ciphers. 

.*» 

PREFIXING  a  cipher   is  placing  it  on  the  left  of  a  number. 

By  prefixing  a  cipher  to  a  decimal,  each  decimal  figure 
is  removed  one  place  to  the  right ;  and  hence,  its  unit  is  di- 

154.  When  is  a  cipher  annexed  to  a  number  ?  Does  the  annexing 
of  ciphers  to  a  decimal  alter  its  value  ?  Why  not  ?  What  do  three- 
tenths  become  by  annexing  a  cipher?  What,  by  annexing  two 
ciphers  ?  Three  ciphers  ?  What  do  8  tenths  become  by  annexing 
a  cipher  ?  By  annexing  two  ciphers  ?  By  annexing  three  ciphers  ? 
What  is  tho  effect  of  placing  a  decimal  point  on  the  right  of  an 
integral  number  and  then  adding  ciphers? 


ADDITION    OF   DECIMALS.  119 

minished  ten  times  (Art.  151)  ;  and  the  same  takes  place  for 
every  cipher  that  is  prefixed  :    Hence, 

Prefixing  ciphers   to   a   decimal  fraction  diminishes  its 
value  ten  times  for  every  cipher  prefixed. 

Take,  for  example,  the  fraction  .2  =  T27. 
.2  becomes     .02    =    ^,     by  prefixing  one  cipher, 
.2  becomes    .002   =  -~~Q,    by  prefixing  two  ciphers, 
.2  becomes  .0002  =  j~0°025,  by  prefixing  three  ciphers: 

in  which  the  fraction  is  diminished  ten  times  for  every  cipher 
prefixed. 


ADDITION    OF    DECIMALS. 

156.  ADDITION  OF  DECIMALS  is  the  operation  of  finding 
the  sum  of  two  or  more  decimal  numbers. 

Only  units  of  the  same  kind  can  be  added  together. 
Therefore,  in  setting  down  decimal  numbers  for  addition,  fig- 
ures having  the  same  unit  value  must  be  placed  in  the  same 
column. 

The  addition  of  decimals  is  then  made  in  the  same  manner 
as  that  of  whole  numbers. 


155.  When  is  a  cipher  prefixed  to  a  number?  When  prefixed 
to  a  decimal,  does  it  increase  the  numerator  ?  Does  it  increase  the 
denominator?  What  effect,  then,  has  it  on  the  value  of  the  frac- 
tion ?  What  does  .2  become  by  prefixing  a  cipher  ?  By  prefixing 
two  ciphers  ?  By  prefixing  three  ?  What  does  .07  become  by  pre- 
fixing a  cipher  ?  By  prefixing  two  ?  By  prefixing  three  ?  By  pre- 
fixing four  ?  What  is  the  effect  of  moving  the  decimal  point  one 
place  to  the  left  ?  Two  places  ?  Three  places  ?  What  is  the  effect 
of  moving  it  one  place  to  the  right  ?  Two  places  ?  Four  places  ? 


120  DECIMAL.  FRACTIONS. 

1.    Find  the  sum  of  37.04,  704.3,  and  .0376. 

OPERATION. 

ANALYSIS. — Place  the  decimal  points  in  the  same  37.04 

column:  this  brings  units  of  the  same  value  in  the         704  3 
same   column:    then   add   as   in   whole   numbers:  0376 

Hence'  "ul^ 

Rule. 

I.  Write  the  numbers  to  be  added,  so  that  figures  of  the 
same  unit  value  shall  stand  in  the  same  column: 

II.  Add  as  in  simple  numbers,  and  point  off  in  the  sum, 
from  the  right  hand,  as  many  places  for  decimals  as  are 
equal  to  the  greatest  number  of  places  in  any  of  the  num- 
bers added. 

PROOF. — The  same  as  in  simple  numbers. 

Examples. 

1.  Add  4.035,  763.196,  445.3741,  and  91.3754  together. 

2.  Add  365.103113,   .76012,   1.34976,    .3549,   and   61.11 
together. 

3.  67.407  +  97.004  +  4  +  .6  +  .06  +  .3. 

4.  .0007  +  1.0436  +  .4  +  .05  +  .047. 

5.  .0049  +  47.0426  +  37.0410  +  360.0039  =  444.0924. 

6.  What  is  the   sum  of  27,   14,  49,  126,  999,  .469,  and 
.2614? 

7.  Add  15,  100,  67,   1,  5,  33,  .467,  and  24.6  together. 

8.  What  is  the  sum  of  99,  99,  31,  .25,  60.102,  .29,  and 
100.347  ? 

9.  Add  together  .7509,  .0074,  69.8408,  and  .6109. 

156.  What  is  Addition  of  Decimals  ?  What  kinds  of  units  may 
be  added  together?  How  do  you  set  down  the  numbers  for  addi- 
tion ?  How  will  the  decimal  points  fall  ?  How  do  you  then  add  ? 
How  many  decimal  places  do  you  point  off  in  the  sum  ? 


ADDITION   OF   DECIMALS.  121 

10.  Required  the  sum  of  twenty-nine  and   3  tenths,  four 
hundred    and    sixty-five,    and    two    hundred    and    twenty-one 
thousandths. 

11.  What  is  the  sum  of  one-tenth,  one-hundredth,  and  one- 
thousandth  ? 

12.  Find  the  sum  of  twenty-five  hundredths,  three  hundred 
and  sixty-five  thousandths,  six-tenths,  and  nine-millionths. 

13.  What  is  the  sum  of  twenty-three  millions  and  ten,  0110 
thousand,   four  hundred  thousandths,   twenty-seven,   nineteen- 
niillionths,  seven,  and  five-tenths  ? 

14.  What  is  the  sum  of  six-millionths,  four  ten-thousandths, 
19  hundred-thousandths,  sixteen-hundredths,  and  four-tenths  ? 

15.  Find  the   sum  of  the   following   numbers :   Sixty-nine 
thousand  and  sixty-nine  thousandths,  forty-seven  hundred  and 
forty-seven  thousandths,  eighty-five  and  eighty-five  hundredths, 
six  hundred   and   forty-nine  and  six  hundred  and  forty-nine 
ten-thousandths. 

16.  A  gentleman  bought  6  houses,  for  which  he  paid,  as 
follows  :    1st,   2785.625    dollars ;    2d,    3964.75    dollars  ;    3d, 
5762.1875  dollars;  4th,  4960.50  dollars;  5th,  6912.375  dol- 
lars; 6th,  9156.3125  dollars:  what  did  the  six  houses  cost? 

17.  A  farmer  sold,  at  different  times,  the  following  quan- 
tities   of    hay :    3.75    tons,    14.165    tons,    375.16247    ton?, 
54.8125    tons,    18.5    tons,    21.75    tons,    and   25    tons :    how 
much  hay  did  he  sell? 

18.  A  vessel  sailed,  in   9  successive  days,   the  following 
distances:  240.17  miles,  315.875  miles,  87.416  miles,  195.125 
miles,  269.1875  miles,  291.06  miles,  197.0106  miles,  300.47925 
miles,  and  200  miles :  what  distance  did  the  vessel  sail  ? 

19.  Add  475.62  ;  nine  hundred   and  twelve  thousandths; 
four  hundred  and  sixty  thousandths  ;   thirty-seven  thousand, 
eight  hundred  and  ninety-nine  ;    one  hundred  and  ninety-nine 
millionths ;  176942.125,  and  two  hundred  and  ninety-six  ten- 
thousandths. 


122  DECIMAL    FRACTIONS. 


SUBTRACTION. 

157.   SUBTRACTION  OF  DECIMALS  is  the  operation  of  finding 
the  difference  between  two  decimal  numbers. 

1.   From  3.275  take  .0879. 

ANALYSIS. — The   subtraction  is  performed  as  in 
whole  numbers,  because  the  units  of  place  in  deci- 
mals have  the   same  relative  values  as  in  whole  .0879 
numbers.                                                                                   3.1871 

In   this   example,   a   cipher   is    annexed   to   the 
minuend,  to  make  the  number  of  decimal  places  equal  to  the 
number  in  the  s.ubtrahend.     This  does  not  alter  the  value  of  the 
minuend  (Art.  154)  :    Hence, 

Rule. 

I.  Write  the  less  number  under  the  greater,  so  that  fig- 
ures of  the  same  unit  value  shall  fall  in  the  same  column. 

II.  Subtract  as  in  simple  numbers,  and  place  the  decimal 
point,  in  the  remainder,  directly  under  that  of  the  subtrahend. 

PROOF. — The  same  as  in  whole  numbers. 

Examples. 

1.  From  3295  take  .0879. 

2.  From  291.10001  take  41.375. 

3.  From  10.000001  take  .111111. 

4.  From  396  take  8  ten-thousandths. 

5.  From  1  take  one-thousandth. 

6.  From  6378  take  one-tenth. 

7.  From  365.0075  take  3  millionths. 

8.  From  21.004  take  97  ten-thousandths. 

9.  From  260.4709  take  47  ten-millionths. 


157.  What  is  Subtraction  of  Decimals  ?  How  do  you  set  down 
the  numbers  for  subtraction?  How  do  you  then  subtract?  How 
many  decimal  places  do  you  point  off  in  the  remainder  ? 


MULTIPLICATION. 


123 


10.  From  10.0302  take  19  millionths. 

11.  From  2.01  take  6  ten-thousandths. 

12.  From  thirty-five  thousand  take  thirty-five  thousandths. 

13.  From  4262.0246  take  23.41653. 

14.  From  346.523120  take  219.691245943. 

15.  From  64.075  take  .195326. 

16.  What  is  the  difference  between  107  and  .0007  ? 

17.  What  is  the  difference  between  1.5  and  .3785? 

18.  From  96.71  take  96.709. 


MULTIPLICATION. 

158.  MULTIPLICATION  OF  DECIMALS  is  the  operation  of 
taking  one  of  two  decimal  numbers  as  many  times  as  there 
are  units  in  the  other. 


OPERATION. 

3.05  =  s     = 


305 
100 


1.    Multiply  3.05  by  4.102. 

ANALYSIS. — We  may  change  the 
factors  into  common  fractions,  and 
then  multiply  them:  the  product  of 
the  numerators  will  be  the  product 
of  the  decimals.  Since  each  denomi- 
nator contains  as  many  ciphers  as 
there  are  places  in  the  numerator 
(Art.  151) ;  and  since  the  product 
of  the  denominators  will  contain  as 
many  ciphers  as  both  the  factors,  it 
follows  that  the  product  of  the  nu- 
merators must  have  as  many  places 
of  figures  as  there  are  in  both  fac- 
tors :  Hence,  the  following 

Rule. 

Multiply  as  in  simple  numbers,  and  point  off  in  the  prod- 
uct, from  the  right  hand,  as  many  figures  for  decimals  as 
there  are  decimal  places  in  both  factors ;  and  if  there  be  not 
so  many  in  the  product,  supply  the  deficiency  by  prefixing 
ciphers. 


4102    1251110 

TWO'    " 

3.05 

4.102 

610 
305 
12.20 

12.51110 


124  /     DECIMAL   FRACTIONS. 

Examples. 

1.  Multiply  the  number  3.049  by  .012. 

2.  Multiply  the  number  365.491  by  .001. 

3.  Multiply  the  number  496.0135  by  1.496. 

4.  Multiply  one  and  one-millionth  by  one-thousandth. 

5.  Multiply  one  hundred  and  forty-seven  millionths  by  one- 
millionth. 

6.  Multiply  three  hundred,  and  twenty-seven   hundredths 
by  31. 

7.  Multiply  31.00467  by  10.03962. 

8.  What  is  the  product  of  five-tenths  by  five-tenths  ? 

9.  "What  is  the  product  of  five-tenths  by  five-thousandths  ? 

10.  Multiply  596.04  by  0.00004. 

11.  Multiply  38049.079  by  0.00008. 

12.  What  will  6.29  weeks'  board  come  to,  at  2.75  dollars 
per  week  ? 

13.  What  will  61  pounds  of  sugar  come  to,  at  0.234  of  a 
dollar  per  pound? 

14.  If   12.83§  dollars  are   paid   for   one   barrel   of  flour, 
what  will  ,354  barrels  cost  ? 

15.  Multiply  49000  by  .0049. 

16.  Bought  1234  oranges  for  4.6  cents  apiece  :  how  much 
did  they  cost  ?  ? 

17.  What  will  375.6  pounds  of  coffee  cost,  at  .125  dollar 
per  pound? 

18.  If  I  buy  36.251  pounds  of  indigo  at  0.029  of  a  dollar 
per  pound,  what  will  it  come  to  ? 

19.  Multiply  $89.3421001  by  .0000028. 

20.  Multiply  $341.45  by  .007. 

21.  What  is  the  product  of  the  decimal  .004  by  the  deci- 
mal .004? 

22.  Multiply  .007853  by  .035. 

23.  What   is    the    product   of  $26.000375   multiplied   by 
.00007? 

158.  What  is  Multiplication  of  Decimals  ?    What  is  the  rule  for 
multiplication  ? 


DIVISION   OF   DECIMALS.  125 


Contractions  in  Multiplication. 

159.  Removing  the  decimal  point  one  place  to  the  right, 
increases  the  unit  of  each  place  ten  times  ;  two  places,  one 
hundred  times,  &c.  Therefore,  when  a  decimal  number  is  to 
be  multiplied  by  10,  100,  1000,  &c.,  the  multiplication  may 
be  made  by  removing  the  decimal  point  as  many  places  to 
the  right  as  there  are  ciphers  in  the  multiplier ;  and  'if  there 
be  not  so  many  figures  on  the  right  of  the  decimal  point, 
supply  the  deficiency  by  annexing  ciphers. 

Examples. 

24.  Multiply  the  number  6.79  by  10  ;   by  100. 

25.  Multiply  the  number  .2694  by  10  ;   by  1000. 

26.  Multiply  the  number  .075  by  100  ;   by  100000. 

27.  Multiply  the  number  1.0049  by  10000000. 


DIVISION. 

160.   DIVISION   OF   DECIMALS    is   the   operation  of  finding 
how  many  times  one  decimal  number  is  contained  in  another. 

1.   Let  it  be  required  to  divide  1.38483  by  60.21. 

ANALYSIS. — The  dividend  must  be  equal 

to  the  product  of  the  divisor  and  quo-  OPERATION. 
tient   (Art.  Y2) ;   and  hence,   must  con-       60.21)1.38483(23 
tain  as  many  decimal  places  as  both  of  1.2042 

them:  therefore,  ~~18063 

There  must  ~be  as  many  decimal  places  18063 

in  the  quotient  as  the  number  of  places 

in   the  dividend  exceeds  the  number  in  Ans.         .023 
the  divisor:    Hence,  the  following 


159.  How  do  you  multiply  a  decimal  number  "by  10,  100,  1000, 
&c.  ?  If  there  are  not  as  many  decimal  figures  as  there  are  ciphers 
in  the  multiplier,  what  do  you  do? 


126  DECIMAL    FRACTIONS. 

Rule. 

Divide  as  in  simple  numbers,  and  point  off  in  the  quo- 
tient, from  the  right  hand,  as  many  places  for  decimals  as 
the  decimal  places  in  the  dividend  exceed  those  in  the  divi- 
sor;  and  if  there  are  not  so  many,  supply  the  deficiency 
V  prefixing  ciphers. 

Examples. 


1.  Divide  2.3421  by  2.11. 

2.  Divide  12.82561  by  3.01. 

3.  Divide  33.66431  by  1.01. 


4.  Divide  .010001  by  .01. 

5.  Divide  8.2470  by  .002. 

6.  Divide  94.0056  by  .08. 


7.  What  is  the  quotient  of  37.57602,  divided  by  3  ;  by  .3 ; 
by  .03  ;  by  .003  ;  by  .0003  ? 

8.  What  is  the  quotient  of  129.75896,  divided  by  8  ;  by 
.08 ;  by  .008  ;  by  .0008 ;  by  .00008  ? 

161.  NOTES. — 1.  When  there  are  more  decimal  places  in  the 
divisor  than  in  the  dividend,  annex  ciphers  to  the  dividend  until 
the  decimal  places  are  equal;  all  the  figures  of  the  quotient  will 
then  be  whole  numbers. 

2.  When  it  is  necessary  to  continue  the  division  further  than 
the  figures  of  the   dividend  will  allow,  we   annex  •  ciphers,  and 
consider  them  as  decimal  places  of  the  dividend. 

When  the  division  does  not  terminate,  we  annex  the  plus  sign 
to  the  quotient,  to  show  that  the  division  may  ba  continued: 
thus,  .2  divided  by  .3  =  .666  +  .  ' 

3.  When  any  decimal   number  is  to   be   divided  by  10,   100, 
1000,  &c.,  the  division  is  made  by  removing  the  decimal  point 
as  many  places  to  the  left  as  there  are  0^  in  the  divisor ;  and  if 
there  be  not  so  many  figures  on  the  left  of  the  decimal  point, 
the  deficiency  is  supplied  by  prefixing  ciphers. 

160.  What  is  Division  of  Decimals?  How  does  the  number  of 
decimal  places  in  the  dividend  compare  with  that  in  the  divisor 
and  quotient  ?  How  do  you  determine  the  number  of  decimal 
places  in  the  quotient  ?  If  the  divisor  contains  four  places  and  the 
dividend  six,  how  many  in  the  quotient  ?  If  the  divisor  contains 
three  places  and  the  dividend  five,  how  many  in  the  quotient  ? 
Give  the  rule  for  the  division  of  decimals  ? 


DIVISION    OF   DECIMALS. 


127 


9. 

Divide 

27.69  by  10. 

16.  Divide 

37.4  by  4.5. 

10. 

Divide 

6.479  by  100 

17.  Divide 

5.864  by  375. 

11. 

Divide 

.056  by  1000 

18.  Divide 

1  by  475.6. 

12. 

Divide 

4397.4  by  3.49. 

19.  Divide 

10  by  12.75. 

13. 

Divide 

.1  by  .0001. 

20.   Divide 

145.764  by  100. 

14. 

Divide 

10  by  .15. 

21.   Divide 

15.64  by  .1. 

15. 

Divide 

.2  by  .6. 

22.  Divide 

16.495  by  1000 

23. 

Divide 

the   number 

2194.02194   by 

.100001. 

24. 

Divide 

the    number 

9811.0047    by 

.325947. 

25. 

Divide 

the   number 

6   by  .6  ;    by  . 

06;   by  .006 

26. 

Divide 

the   number 

6   by  .5;    by  . 

05;    by  .005. 

27. 

Divide 

the   number 

7296.4135    by 

9647.1895. 

28. 

Divide 

the    number 

126.45637    by 

716493.256. 

OPERATION. 
8  )  5.000 

.625 


162.    To  change  a  common  to  a  decimal  fraction. 

The  value  of  a  fraction,  is  the  quotient  of  the  numerator 
divided  by  the  denominator  (Art.  113). 

1.  Reduce  f  to  a  decimal. 

ANALYSIS. — If  we  place  a  decimal  point  after 
the  5,  and  then  write  any  number  of  O's  after  it, 
the  value  of  the  numerator  will  not  be  changed 
(Art.  154). 

If,  then,  we  divide  by  the  denominator,  the 
quotient  will  be  the  decimal  number:  Hence,  the 
following 

Rule. 

Annex  decimal  ciphers  to  the  numerator,  and  then  divide 
by  the  denominator,  pointing  off  as  in  division  of  decimals. 

161.  NOTES. — 1.  If  there  are  more  decimal  places  in  the  divisor 
than  in  the  dividend,  what  do  you  do?    What  will  the  figures  of 
the  quotient  then  be? 

2.  How  do  you  continue  the  division   after  you   have  brought 
down  all  the  figures  of  the  dividend?    What  sign  do  you  place 
after  the  quotient?    What  does  it  show? 

3.  How  do  you  divide  a  decimal  fraction  by  10,  100,  1000,  &c.? 

162.  How  do  you  change  a  common  to  a  decimal  fraction  ?     Is 
the  value  of  the  fraction  altered? 


128  DECIMAL    FRACTIONS. 

Examples. 

1.  Reduce  f  to  a  decimal.     10.  Reduce  4^  to  a  decimal 


2.  Reduce  }f  to  a  decimal.  111.  Reduce 

3.  Reduce  -/-g  to  a  decimal.  !  12.  Reduce  -g-J^. 

4.  Reduce  J  and  T&».          13.   Reduce  -f^. 


5.  Reduce  TVo,  ff>  ToW 


Reduce 


6.  Reduce       and  .        j  lo.  Reduce  —f^. 


7.  Reduce 


8.  Reduce  |,  tfjf, 

9.  Reduce       of      . 


16.  Reduce 

17.  Reduce- 


18. 


347 

25  60' 


10000 


163.     To  change  decimal  to  common  fractions, 

Write  the  denominator  of  the  decimal,  and  reduce  to  the 
lowest  terms. 

Examples. 

1.  Reduce  .04  to  a  common  fraction, 

2.  Reduce  .5,  .25,  and  .125,  to  common  fractions. 

3.  Reduce  4.2,  .875,  and  .375,  to  common  fractions. 

4.  Reduce  3.067  and  8.275  to  common  fractions. 

5.  What  common  fraction  is  equal  to  .00049  ? 

6.  What  common  fractions  are  equal  to  .3125  and  .75? 

7.  What  common  fraction  is  equal  to  .31 1  ? 

8.  What  common  fraction  is  equal  to  .45  f  ? 

Applications  in  the  preceding  Rules. 

1.  What  is  the  sum  of  4J.  7f,  21T\-,  and  12J,  when  ex- 
pressed in  decimals  ? 

2.  Add  6J,  14.375,  14.3125,  l&fe,  and  4627T9T. 

3.  A  merchant  sold  4  parcels  of  cloth  :  the  first  contained 
127  and  3  thousandths  yards  ;   the  second,   6  and  3  tenths 
yards  ;  the  third,  4  and  one-hundredth  yards  ;  the  fourth,  90 
and  one-millionth  yards :  how  many  yards  did  he  sell  in  all  ? 


DIVISION   OF   DECIMALS.  129 

4.  A  merchant  buys  three   chests   of  tea  :    the  first  con- 
tains 60  and  one-thousandth  pounds  ;  the  second,  39  and  -one 
ten-thousandth  pounds  ;  the  third,  26  and  one-tenth  pounds : 
how  much  does  he  buy  in  all  ? 

5.  If  one  man  can  remove  5.91  cubic  yards  of  earth  in  a 
day,  how  much  could  1 9  men  remove  ? 

6.  What  is  the  cost  of  8.3  yards  of  cloth,  at  5.47  dollars 
per  yard? 

7.  What  will  be  the  cost  of  375  thousandths  of  a  cord  of 
wood,  at  2  dollars  a  cord  ? 

8.  A  farmer  sells  to  a  merchant  13.12  cords  of  wood  at 
$4.25   per   cord,    and    13    bushels    of    wheat    at    $1.06   per 
bushel :  he  is  to  take  in  payment  13  yards  of  broadcloth  at 
$4.07  per  yard,  and  the  remainder  in  cash  :  how  much  money 
did  he  receive  ? 

9.  A  gentleman,  having  5456.75  dollars,  not  in  use,  bought 
a  house  for  4896|-  dollars  :  what  was  left  ? 

10.  Ascertain,  by  decimals,  how  much  3j7^  dollars  exceeds 
2J  dollars. 

11.  Multiply,  decimally,  25}  by  74§. 

12.  A  merchant  buys  37|  yards  of  cloth,  at  1.25  dollar 
a  yard  :  what  was  the  cost  of  the  cloth  ? 

13.  What  will  be  the   cost   of  9}   miles   of  railroad,    at 
45675|  dollars  a  mile  ? 

14.  I  gave  28  dollars  to  267  persons :  how  much  was  that 
apiece  ? 

15.  Divide  a  dollar  into  12  equal  parts. 

16.  How  many  times  will  .35  of  35  be  contained  in  .024 
of  24? 

17.  At  .75  of  a  dollar  a  bushel,  how  many  bushels  of  rye 
can  be  bought  for  141  dollars  ? 

18.  Divide  one-millionth  by  one-billionth. 

4  375 

19.  Reduce       '  „  , ,   to  a  decimal  fraction. 

i  of  41 

20.  Reduce  -   — r7-  —  -  to  a  decimal  fraction. 

4g 

6*  •  Sfr 


130  UNITED   STATES   CURRENCY. 


UNITED    STATES    CURRENCY. 

164.  In  the  year  1792,  Congress  established  the  Decimal 
Currency,  as  the  Currency  of  the  United  States. 

The  unit  of  this  currency,  is  1  dollar,  denoted  by  $1  ; 
10  dollars  make  1  Eagle ;  one-tenth  of  a  dollar,  1  dime ; 
one-tenth  of  a  dime,  or  one-hundredth  of  a  dollar,  1  cent ; 
one-tenth  of  a  cent,  or  one-thousandth  of  a  dollar,  1  mill: 
as  shown  in  the  following 


Table. 

10  Mills  (m.) 
10  Cents    .     . 
10  Dimes  .     .- 
10  Dollars      . 

.     make     1  Cent, 
1  Dime, 
1  Dollar, 
1  Eagle, 

.     marked  ct. 
11       d. 
"       $. 
"       E. 

Table  Reversed. 

Eagles.    Dollars.       Dimes.  Cents.  Mills. 

1  =  10. 

1   =       10  =  100. 

1   =     10  =     100  =  1000. 

1   =   10  =   100  =   1000  =  10000. 

Dimes  are  generally  read  in  cents:   thus,  4  dimes  is 
read,  40  cents;  and  4  dimes  and  6  cents,  46  cents. 

Coins. 

165.  COINS,  are  pieces  of  metal,  whose  values  are  fixed 
by  law. 

The  corns  of  the  United  States  are  of  gold,  silver,  copper, 
and  nickel,  and  are  of  the  following  denominations : 

1.  Gold  :     Eagle,    double-eagle,    half-eagle,    three-dollars, 
quarter-eagle,  dollar. 

2.  Silver :    Dollar,    half-dollar,    quarter-dollar,    dime,   half- 
dime,  and  three-cent  piece. 

3.  Copper:    Cent,  half-c<5nt. 
4;   Nickel:   Cent. 


REDUCTION.  131 

Expressing  Currency  decimally. 

1.  Express  $89  and  39  cents  and  7  mills,  decimally. 

2.  Express  $12  and  3  mills,  decimally. 

3.  Express  $147  and  4  cents,  decimally. 

4.  Express  $148,  4  mills,  decimally. 

5.  Express  $4,  6  mills,  decimally. 

6.  Express  $9,  6  cents,  9  mills,  decimally. 

7.  Express  $10,  13 -cents,  2  mills,  decimally. 

8.  Express  25  cents,  decimally. 

9.  Express  18  J  cents,  decimally. 

10.  Express  4  cents  and  7  mills,  decimally. 

11.  Read  in  dollars,  dimes,  cents,  and  mills,  $24.185. 

12.  Read  $135.3125  ;  $607.437,  and  $0.634. 

Reduction. 

166.  REDUCTION  is  the  operation  of  changing  the  unit  of 
a  number,  without  altering  the  value  of  the  number.  We 
see,  from  the  foregoing  Table,  that  1  unit  of  any  denomina- 
tion is  equal  to  10  units  of  the  next  iower. 

167.    To  change  from  a  greater  unit  to  a  less. 

1.  To  change  from  any  denomination  to   the  next  lower, 
multiply  by  10. 

2.  To  change  from  any  denomination  to  the  second  lower, 
multiply  by  100. 

3.  To  change  from  any  denomination  to  the  third  lower, 
multiply  by  1000. 

NOTE. — If  there  be  no  decimal  point  in  the  number,  perform 
the  operation  by  annexing  ciphers.  If  there  is  a  decimal  point, 

164.  What  is  the  Currency  of  the  United  States?    What  is  the 
unit  of  United   States  Currency  ?  s  What  are  its   denominations  ? 
Give  the  Table. 

165.  What  are  coins  ?    What  metals  are  used  in  the  coins  of  the 
United   States?    What   are   the   gold   coins?     What   the   silver? 
Copper  ? 

166.  What  is  Reduction? 


132  UNITED   STATES  CURRENCY. 

remove  it  as  many  places  to  the  right  as  there  are  O's  in  the 
multiplier;  and  omit  the  $  mark,  when  the  answer  is  in  dimes, 
cents,  or  mills. 

Examples. 

1.  How  many  dimes,  in  $56?     In  $96.47?     In  $1.06? 

2.  How  many  cents,  in  $16?     In  $45.625?     In  $1.0875? 

3.  How  many  mills,  in  $4.156  ?     In  $69  ?     In  $0.75  ? 

168.    To  change  from  a  less  unit  to  a  greater. 

1.  To  change  from  any  denomination  to  the  next  greater, 
divide  by  10. 

2.  To  change  from  any  denomination  to  the  second  next 
higher,  divide  by  100. 

3.  To  change   from   any  denomination  to  the   third  next 
higher,  divide  by  1000. 

Examples. 

1.  Reduce  45689  mills  to  dollars. 

2.  In  6794  cents,  how  many  dollars  ?     How  many  dimes  ? 

3.  How  many  dollass  are  there  in  376594  cents  ? 

4.  How  many  cents  in  47546  mills  ? 

5.  How  many  Eagles  in  67506  mills  ? 

6.  How  many  dollars  in  37496  mills  ? 

7.  How  many  dollars  in  47049  mills  ? 

8.  In  8756  cents,  how  many  dimes?     How  many  dollars ? 


ADDITION   OF   CURRENCY. 

169.  Since  Decimal  Currency  follows  the  decimal  system, 
it  may  be  added,  subtracted,  multiplied,  and  divided  by  the 
rules  of  decimals. 


167.  How  do  you  change  a  greater  unit  to  a  less  ? 

168.  How  do  you  change  a  less  unit  to  a  greater  ? 


ADDITION.  133 

1.  John  gives  $1.37|  for  a  pair  of  shoes,  25  cents  for  a 
penknife,  and  1 2£  cents  for  a  pencil :  how  much  does  he  pay 
for  all  ? 

ANALYSIS. — We  place  the  numbers  so  that  units  ON* 

of  the  same  value  may  be  in  the   same   column, 
having  reduced   the  half-cents   to  decimals.     The  c 

addition  is  performed  as  in  addition  of  decimals : 
Hence,  $1.750 

Rule. 

Write  the  numbers  so  that  units  of  the  same  value  shall 
fall  in  the  same  column,  and  then  add  as  in  decimal 
fractions. 

PROOF. — Same  as  in  decimals. 

Examples. 

1.  Add  $0.047,  $6.210,  $0.47,  and  3  mills. 

2.  Add  4  dollars,  16  cents,  87  cents,  and  95  mills. 

3.  Add  120  dimes,  374  cents,  and  956  mills. 

4.  Add  $67.214,  $10.049,  $6.041,  $0.271,  together. 

5.  Add  $59.316,  $87.425,  $48.872,  $56.708,  together. 

6.  Add  $81.053,  $67.412,  $95.376,  $87.064,  together. 

Applications. 

1.  A  grocer  purchased  a  box  of  candles,  for  6  dollars  89 
cents ;  a  box  of  cheese,  for  25  dollars  4  cents  and  3  mills ; 
a  keg  of  raisins,  for  1  dollar  12£  cents   (or  12  cents  and  5 
mills) ;  and  a  cask  of  wine,  for  40  dollars  37  cents  8  mills : 
what  did  the  whole  cost  him  ? 

2.  A  farmer  purchased  a  cow,  for  which  he  paid  30  dol- 
lars and  4  mills ;  a  horse,  for  which  he  paid  104  dollars  60 
cents   and  1  mill;   a  wagon,  for  which   he   paid   85   dollars 
and  9  mills :  how  much  did  the  whole  cost  ? 

169.  How  do  you  set  down  the  numbers  for  addition  ?  How  do 
you  add  up  the  columns  ?  How  do  you  place  the  separating  point  1 
How  do  you  prove  addition  ? 


134:  UNITED   STATES   CURRENCY. 

3.  Mr.  Jones    sold    farmer   Sykes    6    chests    of    tea.    for 
$75.641 ;    9  yards   of  broadcloth,    for  $27.41  ;   a   plow,  for 
$9.75,  and  a  harness,  for  $19.674 :  what  was  the  amount  of 
the  bill  ? 

4.  A  grocer  sold  Mrs.  Williams  18  hams,  for  $26.497  ;  a 
bag  of  coffee,  for  $17.419;  a  chest  of  tea,  for  $27.047,  and 
a  firkin  of  butter,   for  $28.147  :   what  was  the   amount  of 
her  bill? 

5.  A  father  bought  a  suit  of  clothes  for  each  of  his  four 
boys  :    the  suit  of  the  eldest   cost   $15.167 ;   of  the  second, 
$13.407  ;  of  the  third,  $12.75,  and  of  the  youngest,  $11.047: 
how  much  did  he  pay  in  all  ? 

6.  A  father  has  six  children :   to  the  first  two  he   gives 
each  $375.416  dollars;  to  each  of  the  second  two,  $287.55; 
to  each  of  the  third  two,  $259.004  :  how  much  does  he  give 
to  them  all? 

7.  A   man   is   indebted  to    A,   $630.49 ;    to   B,   $25 ;    to 
C,  87|  cents ;  to  D,  4  mills :  how  much  does  he  owe  ? 

8.  Bought    1    gallon   of  molasses,   at   28   cents  ;    a   half- 
pound  of  tea,  for  78  cents ;  a  piece  of  flannel,  for  12  dollars 

6  cents  and  3  mills ;  a  plow,  for  8  dollars  1  cent  and  1  mill ; 
and  a  pair  of  shoes,  for  1   dollar   and  20   cents :    what   did 
the  whole  cost  ? 

9.  Bought  6  pounds  of  coffee,  for  1  dollar  12|  cents ;   a 
wash-tub,  for  75  cents  6  mills ;  a  tray,  for  26  cents  9  mills ; 
a  broom,  for  27  cents ;  a  box  of  soap,  for  2  dollars  65  cents 

7  mills  ;  a  cheese,  for  2  dollars  87^  cents :  what  is  the  whole 
amount  ? 

10.  What  is  the  entire  cost  of  the  following  articles,  viz.: 
2  gallons   of  molasses,  57  cents  ;  half  a  pound  of  tea,  37 i 
cents ;  2  yards  of  broadcloth,  $3.37|  cents ;  8  yards  of  flan- 
nel, $9.875  ;  two  skeins   of  silk,  12J  cents,  and  4  sticks  of 
twist,  8 1  cents  ? 

11.  A  person  paid  $569.75   for  muslin,  $1256|-  for  silk, 
$674.12|  for  calico,  $962 j  for  flannel,  $169.1875  for  thread; 
what  did  he  pay  for  all  ? 


SUBTRACTION   OF   CURRENCY.  135 


SUBTRACTION. 

170.  1.  A  man  buys  a  cow  for  126.37,  and  a  calf  for 
$4.50 :  how  much  more  does  he  pay  for  the  cow  than  for 
the  calf? 

ANALYSIS. — The  operation  is  performed  as  in  OPERATION. 
subtraction  of  decimals.  $26.37 

4.50 

RULE. — Same  as  in  Subtraction  of  Decimals.  .        ^ 

•al.ol 
PROOF. — The  same  as  m  Subtraction. 

Examples. 

(1.)  (2.) 

From        $204.679  From        $8976.400 

Take  98.714  Take  610.098 

Remainder,  Remainder, 

.    (3.)  (4.)  (5.) 

$620.000  $327.001  $2349 

19.021  2.090  29.33 


6.  What  is  the  difference  between  $6  and  1  mill?    Be- 
tween $9.75  and  8  mills  ?     Between  75  cents  and  6  mills  ? 
Between  $87.354  and  9  mills? 

7.  From  $107.003   take   $0.479. 

8.  From   $875.043   take   $704.987. 

9.  From   $904.273   take   $859.896. 

10.  A  man's  income  is  $3000  a  year ;  he  spends  $487.50 : 
how  much  does  he  lay  up? 

11.  A  man  purchased  a  yoke  of  oxen  for  $78,  and  a  cow 
for  $26.003 :  how  much  more  did  he  pay  for  the  oxen  than 
for  the  cow? 

12.  A  man  buys  a  horse  for  $97.50,  and  gives  a  hundred- 
dollar  bill :  how  much  ought  he  to  receive  back  ? 

13.  JIow  much  must  be  added  to  $60.039,  to  make  the 
sum  $1005.40  ? 


136  UNITED   STATES   CURRENCY. 

14.  A   man   sold   his   house    for   $3005,   this   sum   being 
$98.039  more  than  he  gave  for  it :  what  did  it  cost  him  ? 

15.  A  man  bought   a  pair  of  oxen   for  $100,   and   sold 
them   again  for  $75.3 7 £ :  did  he  make  or  lose  by  the  bar- 
gain, and  how  much  ? 

16.  A  man   starts   on    a  journey  with   $100 ;   he   spend 
$87.57  :   how  much  has  he  left  ? 

17.  How  much  must  you  add  to  $40.173,  to  make  $100? 

18.  A  merchant  stocked  a  store,  at  the  beginning   of  a 
year,  at  an  expense  of  $10500  ;  during  the  year  he  bought 
goods  to  the  amount  of  $9475f,  and  sold  to  the  amount  of 
$14500.625;  his  expenses  were  $12 67.31  J;  and  the  goods  in 
the  store,  at  the   close  of  the  year,  are  worth    $8965.459  : 
what  was  his  gain  during  the  year? 

19.  A  farmer  had   a   horse   worth   $147.49.     He   traded 
him  for  a  colt  worth  $35.048,  and  a  cow  worth  $47T\,  and 
received  the  rest  in  cash  :  what  cash  did  he  receive  ? 

20.  My  house  is  worth  $8975.034  ;  my  barn,  $695.879 : 
what  is   the   difference   of  their  values,   and  what  would  be 
gained  if  both  were  sold  for  $10000  ? 

21.  What  is  the  difference  between  nine  hundred  and  sixty- 
nine  dollars   eighty  cents   and   1   mill,   and  thirty-six   dollars 
ninety-nine  cents  and  9  mills  ? 


MULTIPLICATION. 

171.     1.   A   farmer   sells    8   sheep   for   $1.25   each  :    how 
much  does  he  receive  for  the  whole  ? 

ANALYSIS. — The  price   of  one  sheep  Is  multiplied  OPERATION. 
by  the  number  of  sheep.     The  multiplicand  is  the      $   1.25 
price  of  one,  in  decimals;  and  the  multiplier  is  an  8 

abstract    number,    denoting    the    number    of  things      *VrTnn 
whose  value  is  required.    The  product,  however,  will 
be  the  same,  whichever  number  is  used  as  the  multiplier  (Art 
61):    Hence, 


MULTIPLICATION.  137 

Rule. 

Multiply  and  point  off,  as  in  decimal  fractions. 

Examples. 

1.  What  will  8  barrels  of  flour  cost,  at  $6.375  per  barrel  ? 

2.  What  will  55  yards  of  cloth  come  to,  at  37|  cents  per 
yard  ? 

3.  What  will  300  bushels  of  wheat  come  to,  at  $1.25  per 
bushel  ? 

4.  What  will   85.25   pounds   of  tea  cost,   at   $1.37|   per 
pound  ? 

5.  What  is   the  cost  of  a  cask  of  wine  containing  29| 
gallons,  at  $2.75  per  gallon? 

6.  What  will  be  the  cost  of  47  barrels  of  apples,  at  $1J 
per  barrel  ? 

7.  What  is  the  cost  of  a  box  of  oranges,  containing  450, 
at  2J  cents  apiece  ? 

8.  What  is  the  cost  of  307  yards  of  linen,  at  68£  cents 
per  yard  ? 

9..  If  1  pound  of  butter  cost  12  J  cents,  what  will  4  firkins 
cost,  each  weighing  56  pounds? 

10.  At  $1.33J  a  foot,  what  will  it  cost  to  dig  a  well  78 
feet  deep  ? 

11.  If  it  cost  $2.3125  to  keep  a  horse  for  1  week,  what 
would  be  the  cost  of  keeping  3  horses  for  18-f  weeks  ? 

12.  A  flour  merchant  bought  125  barrels  of  flour  at  $5J- 
each,  and  sold  them  at  $5.9375  :   what  was  the  amount  of 
gain? 

13.  A  speculator  sold  16  houses  at  $2463. 12|  each,  and 
received  in  exchange  25  lots  of  ground,  each  worth,   on  an 
average,    $872J,   and  the  rest   in   cash :    what  cash   did   he 
receive  ? 

14.  A  cloak  required  6|  yards  of  cloth.    What  would  be 
the  cost  of  the  cloak,  if  the  cloth  cost  $3f  a  yard,  and  as 
many  yards  of  lining  at  $0.98|  a  yard,  and  the  making  cost 
$3.45  ? 


138  UNITED   STATES   CURRENCY. 


DIVISION. 

172.     1.   Bought  9  pounds  of  tea  for  $5.85  :    what  was 
the  price  per  pound? 

ANALYSIS. — One  pound  will  cost  ^  as  much  as    OPERATION. 
9  pounds.    |  of  $5.85  is  found  by  dividing  by  9,      9  )  5.85 
according  to  the  principles  of  Division  of  Deci-  77 

mals :    Hence  the 

Rule. 

Arrange  the  numbers  for  division,   and  proceed  as  in 
division  of  decimals. 

PROOF. — The  same  as  in  division  of  decimals. 


Examples. 


1.  Divide  $56.16  by  16. 

2.  Divide  $495.704  by  129. 

3.  Divide  $12  by  200. 


4.  What  is  ^  of  $400  ? 

5.  What  is  ^r  of  $857  ? 

6.  What  is  ffo  of  $6578.95  ? 

7.  Paid  $29.68  for  14  barrels  of  apples  :    what  was  the 
price  per  barrel  ? 

8.  If  27  bushels   of  potatoes   cost  $10.125,  what  is  the 
price  of  a  bushel  ? 

9.  If  a  man  receive  $29.25  for  a  month's  work,  how  much 
is  that  a  day,  allowing  26  working  days  to  the  month? 

10.  A  produce-dealer  bought  3  barrels  of  eggs,  each  con- 
taining 150   dozen,  for  which  he  paid  $63  :   how  much  did 
he  pay  a  dozen  ? 

11.  A  man  bought  a  piece  of  cloth  containing  72  yards, 
for  which  he  paid  $252  :    what  did  he  pay  per  yard  ? 

12.  If  $600  be  equally  divided  among   26   persons,  what 
will  be  each  one's  share  ? 

13.  A  lady  bought  17|  yards   of  silk  for  $15.75  :    what 
was  the  price  per  yard  ? 

172.  What  is  the  rule  for  division  of  United  States  Money? 


APPLICATIONS.  139 

14.  If  $47.31J   paid   for   the  board    of   a   family   for  5£ 
weeks,  what  was  the  price  per  week? 

15.  If  I  pay  $4.50  a  ton  for  coal,  how  much  can  I  buy 
for  $67.50? 

16.  At  $7  a  barrel,  how  much  flour  can  be  bought  for 
8178.50? 

17.  How  many  pounds  of  tea  can  be  bought  for  $6.75,  at 
75  cents  a  pound  ? 

18.  What  number  of  barrels  of  apples  can  be  bought  for 
$47.50,  at  $2.37J  a  barrel  ? 

19.  At  44  cents  a  bushel,  how  many  bushels  of  oatfa  can 
be  bought  for  $14.30  ? 

20.  At  34  cents  a  bushel,  how  many  barrels  of  apples  can 
I  buy  for  $13.60,  allowing  2|  bushels  to  the  barrel? 

21.  A  farmer  receives  $840  for  the  wool  of  1400  sheep  : 
how  much  does  each  sheep  produce  him  ? 

22.  A  merchant   buys   a   piece   of  goods   containing    105 
yards,  for  which  he  pays  $262.50  ;   he  wishes   to  sell  it   so 
as  to  make  $52.50  :   how  much  must  he  ask  per  yard  ? 

23.  If   1    acre    of   land   cost   $28.75,   how   much   can  be 
bought  for  $3220  ? 

24.  Paid  $40.50  for  a  pile  of  wood,  at  the  rate  of  $3.37J 
a  cord,  how  much  was  there  in  the  pile  ? 

Applications. 

173.  An  ALIQUOT  PART  of  a  number  is  any  exact  divisor, 
whether  fractional  or  integral.  Thus,  3  months  is  an  aliquot 
part  of  a  year,  being  one-fourth  of  it ;  and  12£  cents  is  an 
aliquot  part  of  1  dollar,  being  one-eighth  of  it. 

Aliquot  Parts  of  a  Dollar. 


$1  =100    cents. 

J  of  a  dollar  =  50    cents. 

|  of  a  dollar  =  33  J  cents. 

J  of  a  dollar  =25    cents. 

j  of  a  dollar  =20    cents. 


of  a  dollar  =  12|  cents, 
of  a  dollar  =10  cents, 
of  a  dollar  =  6  J  cents, 
of  a  dollar  =  5  cents, 
of  a  cent  =  5  mills. 


140  UNITED   STATES   CURRENCY. 

CASE     I. 

174.  To  find  the   cost  of  several  things,  when  the  price 
of  a  single  thing  is  an  aliquot  part  of  $1. 

1.  What  will  be  the  cost  of  69  yards  of  cotton  sheeting, 
at  33|  cents  a  yard  ? 

ANALYSIS.— 33^-  cents  =  $  of  a  dollar :  69  yards,       OPERATION. 
at  $1  a  yard,  would  cost  $69 :  at  £  of  a  dollar  a  3  )  69 

vard,  it  will  cost  i  of  $69,  which  is  $23 :  Hence,  |23 

Rule. — Take  such  a  part  of  the  number  of  things,  as 
the  price  of  a  single  thing  is  of  $1. 

Examples. 

1.  What  will  100  cocoa-nuts  cost,  at  12lcts.=-|-  of  a  dollar 
apiece  ? 

2.  What  will  250  water-melons  cost,  at  25  cents  apiece  ? 

3.  What  will  55  tops  cost,  at  6|  cents  each? 

4.  What  will  650  yards   of  muslin   cost,   at  12 J  cents  a 
yard  ? 

5.  What  will  450  melons  cost,  at  5  cents  apiece  ? 

6.  What  will  6640  lemons  cost,  at  6J  cents  each  ? 

7.  What  will  136  gallons  of  molasses  cost,  at  33|  cents 
a  gallon  ? 

CASE     II. 

175.  To  find  the   cost,   -when  the  price  of  one,  and  the 
number  of  things  are  given. 

1.    What  is  the  cost  of  36  oranges,  at  5  cents  apiece  ? 

OPERATION. 

ANALYSIS. — Since  5  cents  taken  36  times,  is  the  Op 

same  as  36  taken  5  times  (Art.  51),  the  cost  will  be 
the  product  of  5  by  36,  or  of  36  by  5 :     Hence, 

$1.80 

Rule. — Multiply  the  price  of  I  by  the  number  of  things, 
or  the  number  of  things  by  the  price  of  1,  and  the  product 
will  be  the  cost. 


APPLICATIONS.  141 

Examples. 

1.  What  will  367  hats  cost,  at  $1.12£  each  ? 

2.  What  will   2479    bushels    of  wheat   cost,    at   $1.50    a 
bushel  ? 

3.  What  will  4204  yards  of  cloth  cost,  at  $2.37|  a  yard? 

4.  What  will  3270  barrels  of  flour  cost,  at  $8.45  a  barrel  ? 

5.  What  will  be  the  cost  of  3204  cheeses,  at  $2.39  apiece  ? 

6.  What  will  be  the  cost  of  694  sheep,  at  $4.29  apiece? 

CASE     III. 

176.  To  find  the  cost  of  things  sold  by  the  hundred  or 
thousand. 

1.  What  will  be  the  cost  of  936  feet  of  lumber,  at  3 
dollars  per  hundred  ? 

ANALYSIS. — At  3  dollars  a  foot,  the  cost  would  be  OPERATION. 
936  x  3  —  2808  dollars;  but  as  3  dollars  is  the  price  936 

of  100  feet,  2808  dollars  is  100  times  the  cost  of  3 

the  lumber :  therefore,  if  we  divide  2808  dollars  by        ^Q  ng 
100,  which  we  do  by  cutting  off  two  of  the  right- 
hand  figures  (Art.  77),  we  shall  obtain  the  cost. 

NOTE. — Had  the  price  been  so  much  per  thousand,  we  should 
have  divided  by  1000,  or  cut  off  three  of  the  right-hand  figures. 

Rule. — Multiply  the  number  and  price  together,  and 
point  off  in  the  product,  two  places  of  decimals  more  than 
there  are  in  both  factors,  when  sold  by  the  hundred,  and 
three  places  more,  when  sold  by  the  thousand. 

Examples. 

1.  What  will  4280  bricks  cost,  at  $5  per  1000? 

2.  What  will  2673  feet  of  timber  cost,  at  $2.25  per  C  ? 

3.  What  will  be  the  cost  of  576  feet  of  boards,  at  $10.62 
per  M? 


174.  What  is  Case  I.?    Give  the  rule. 

175.  What  is  Case  II.  ?    Give  t  he  rule. 

176.  What  is  Cage  III.?    Give  the  rule. 


142  UNITED   STATES   CURRENCY. 

4.  What  is  the  value  of  1200  feet  of  lathing,  at  7  dollars 
per  M? 

5.  What  will  be  the  freight  of  6727  pounds,  from  Buffalo 
to  New  York,  at  $2.45  per  100  pounds  ? 

6.  What  will  27097  pounds  of  butter  cost,  at  $28J  per  C  ? 

CASE    IV. 

177.  To  find  the  cost  of  articles  sold  by  the  ton  of  2000 
pounds,  when  the  price  of  a  ton  is  known. 

1.   What  is   the  cost  of  1684  pounds  of  hay,  at  $10.50 
per  ton  ? 

OPEKATION. 

ANALYSIS. — The  cost  of  1000  pounds  will  ~  \  i  n  50 
be   one-half  the    cost   of   2000    pounds,   or 
$5.25.      Then,    by    the    last    case,    if   1000  5.25 

pounds   cost   $5.25,    1684   pounds   will    cost  1684 

$8.84100.  $8.84100,  Ans. 

Rule. — Divide  the  price  by  2,  and  then  find  the  cost 
of  the  quantity  by  the  last  Case. 

Examples. 

1.  What  will  3426  pounds  of  plaster  cost,  at  $3.48  per 
ton? 

2.  What  is  the  cost  of  the  transportation  of  6742  pounds 
of  iron  from  Kuffalo  to  New  York,  at  7  dollars  per  ton  ? 

3.  What  is  the  cost  of  6527  pounds  of  oats,  at  $30.25 
per  ton  ? 

4.  What  is  the  cost  of  1678  pounds  of  coal,  at  $8.75 
per  ton  ? 

5.  What  is  the  transportation  of  37941   pounds  of  rail- 
road iron  from  New  York  to  Buffalo,  at  $7.37^  per  ton  ? 

CASE    V. 

178.  When  the   number   of  things    is    known,    and  their 
cost:   to  find  the  price  of  1  thing. 


177.  What  is  Case  IV.  ?    Give  the  rule. 


APPLICATIONS.  143 

1.   If  7  pounds  of  tea  cost  $4.55  :  what  is  the  price  per 
pound  ? 

ANALYSIS. — 1  pound  will  cost  one-seventh  as          OPERATION. 
much  as  7  pounds:  one-seventh  of  $4.55  is  65  7  )  4.55 

cents;  therefore,  1  pound  will  cost  65  cents.  AQ  gc 

Rule. — Divide  the  entire  cost  by  the  number  of  things. 

Examples. 

1.  Divide  $3769.25  into  50  equal  parts  :  what  is  one  part? 

2.  A  farmer  purchased  a  farm   containing  725  acres,  for 
which  he  paid  $18306.25:  what  did  it  cost  him  per  acre? 

3.  A  merchant   buys    15   bales   of  goods   at   auction,  for 
which  he  pays  $1000:  what  do  they  cost  him  per  bale? 

4.  A  drover   pays   $1250  for  500   sheep  :   what  shall  he 
sell  them  for  apiece,  that  he  may  neither  make  nor  lose  by 
the  bargain  ? 

CASE    VI. 

179.    "When  the  cost  of  a  number  of  things  is  given,  and 
the  price  of  1 :  to  find  the  number. 

1.    If  I  pay  84.50  a  ton  for  coal,  how  much  can  I  buy 
for  $67.50  ? 

ANALYSIS.— As  many  tons  as  $4.50  OPUIATION. 

is   contained   times   in   $67.50,  which     4.50)67.50(15  tons. 
is   15.  67.50 

Rule. — Divide  the  entire  cost  by  the  cost  of  I  thing. 

Examples. 

1.  If  1  acre  of  land  cost  $38.75,  how  much  can  be  bought 
for  $3560  ? 

2.  How  many  sheep  can  be  bought  for  $132,  at  1.37J  a 
head  ? 

3.  At  $4.25  a  yard,   how  many     yards  of  cloth  can  be 
bought  for  $68  ? 

178.  What  is  Case  V.  ?    Give  the  rule. 

179.  What  is  Case  VI.  ?    Give  tho  rule. 


144  UNITED   STATES   CURRENCY. 


Miscellaneous  Examples. 

1.  If  12  tons  of  hay  cost  $150,  what  will  50  tons  cost? 

2.  If  9  dozen  spelling-books  cost  $7.875,  what  will  6  dozen 
cost?     8  dozen?     15|  dozen? 

3.  If  75 1  bushels  of  wheat  cost  $131.25,  how  much  will 
9J  bushels  cost?     37.375  bushels? 

4.  If  320  pounds  of  coffee  cost  $44.80,   how  much   will 
575  pounds  cost  ? 

5.  Mr.  James  B.  Smith  bought  9  barrels  of  sugar,  each 
weighing  216  pounds,  for  which  he  paid  $116.64 :  how  much 
did  he  pay  a  pound? 

6.  Mr.  Wilson  spent  T5g-  of  120  dollars  for  his  wood,  and 
the  remainder  for  coal :  the  wood  was  worth  $3.75  per  cord, 
and  the  coal  $4.3  TJ   per  ton:   what  quantity  of  each   was 
bought  ? 

7.  A  gentleman  bought  1  pound  of  coffee  for  $.18  j  ;  one 
pound  of  tea,  for  $.87|  ;    one  pound   of  butter,  for  $.31 J  ; 
and  one  bushel  of  potatoes,  for  $T7g-.    He  gave  a  two-dollar 
bill :  what  change  must  he  receive  ? 

8.  A  farmer  sold  a  yoke   of  oxen   for  $80.75  ;    6  cows, 
for  $29  each  ;  30  sheep,  at  $2.50  a  head  ;  and  3  colts,  one 
for  $25,  the  other  two  for  $30  apiece :  what  did  he  receive 
for  the  whole  lot  ? 

9.  A   merchant   buys   6   bales  of  goods,   each   containing 
20  pieces  of  broadcloth,  and  each  piece  of  broadcloth  con- 
tained  29  yards :    the  whole   cost  him   $15660 :   how  many 
yards  of  cloth  did  he  purchase,  and  how  much  did  it  cost 
him  per  yard  ? 

10.  A  person  sells  3  cows,  at  $25   each  ;    and  a  yoke  of 
oxen  for  $65  :  he  agrees  to  take  in  payment  60  sheep :  how 
much  do  his  sheep  cost  him  per  head? 

11.  A   man    dies,    leaving    an    estate    of   $33000,    to    be 
equally  divided   among  his   4   children,    after  his  wife   shall 
have   taken   her  third.     What  was   the  wife's  portion,   and 
what  the  part  of  each  child  ? 

12.  A  person  settling  with  his  butcher,  finds  that   he   is 


APPLICATIONS.  H5 

charged  with  126  pounds  of  beef,  at  9  cents  per  pound ;  85 
pounds  of  veal,  at  6  cents  per  pound ;  6  pair  of  fowls,  at 
37  cents  a  pair,  and  three  hams,  at  $1.50  each:  how  much 
does  he  owe  him  ? 

13.  A  farmer  agrees  to  furnish  a  merchant  40  bushels  of 
rye,  at  62  cents  per  bushel,  and  to  take  his  pay  in  coffee, 
at  16  cents  per  pound:  how  much  coffee  will  he  receive? 

14.  A   farmer  has   6   ten-acre  lots,   in   each  of  which  he 
pastures  6  cows  ;  each  cow  produces  112  pounds  of  butter, 
for  which  he  receives  18|  cents  per  pound;  the  expenses  of 
each  cow  are  5  dollars  and  a  half :  how  much  does  he  make 
by  his  dairy? 

15.  Bought  a  farm  of  W.  N.  Smith,  for  2345  dollars ;  a 
span  of  horses,  for  375  dollars ;   6  cows,  at  36  dollars  each. 
I   paid   him  520   dollars   in  cash,   and  a  village  lot,  worth 
1500  dollars :  how  many  dollars  remain  unpaid  ? 

16.  Divide  .5  by  .5  ;   12J  by  |  ;    81 J  by  .8125. 

17.  A  man  leaves  an  estate  of  $1473.194,  to  be  equally 
divided  among  12  heirs  :    what  is  each  one's  portion? 

18.  If  flour  is  $9.25  a  barrel,  how  many  barrels  can  I  buy 
for  $1637.25? 

19.  Bought  26  yards  of  cloth  at  $4.37£  a  yard,  and  paid 
for  it  in  flour  at  $7.25  a  barrel  :  how  much  flour  will  pay 
for  the  cloth? 

20.  How  much  molasses,  at  22|  cents  a  gallon,  must  be 
given  for  46  bushels  of  oats,  at  45  cents  a  bushel  ? 

21.  How  many  days'  work,  at  $1.25  a  day,  must  be  given 
for  6  cords  of  wood,  worth  $4.12J  a  cord? 

22.  David  Trusty,    '  Bought  of  Peter  Big  tree, 

2462  Feet  of  boards,  at  $7  per  M.     . 

4520     "              "  "  9.50        " 

600     "         scantling,  "  11.37        "         .    /    . 

900     "         timber,  "  15,            " 

1464    "         lathing,  "  .75  per  C.     . 

1012     "'        plank,  "  1.25        " 

Received  payment, 

PETEB  BIGTREE. 

7 


H6  UNITED    STATES    CURKENCY. 

23.  NEW  YORK,  May  1,  1862. 

Mr.  James  Spendthrift, 

Bought  of  Eenj.  Saveall, 

16  Pounds  of  tea,  at  $.85  per  pound  .... 

27       "        "  coffee,  at  $.15|  per  pound     . 

15  Yards  of  linen,  at  $.66  per  yard    ....  


Amount,        .        .        .  $ 
Received  payment, 

BENJAMIN  SAVEALL. 


24.  ALBANY,  June  2,  1862. 

Mr.  Jacob  Johns, 

Sought  of  Gideon  Gould, 

36  Pounds  of  sugar,  at  9|-  cents  per  pound 
8  Hogsheads  of  molasses,  63  galls,  each,  at  27  cents 

a  gallon      .        .        .        .        . 
5  Casks  of  rice,  285  pounds  each,  at  5  cts.  per  pound 
2  Chests  of  tea,  86  pounds  each,  at  96  cts.  per  pound 

Total  cost,        .        .        $ 
Received  payment,  for  GIDEON  GOULD, 

CHARLES  CLARK. 


25.  HABTFOBD,  November  21,  1802 

Gideon  Jones, 

Sought  of  Jacob  Thrifty, 

69  Chests  of  tea,  at  $55.65  per  chest 
126  Bags  of  coffee,  100  pounds  each,  at  12i  cents  per 
pound      ........ 

167  Boxes  of  raisins,  at  $2.75  per  box 
800  Bags  of  almonds,  at  $18.50  per  bag    . 
9004- Barrels  of  shad,  at  $7.50  per  barrel    . 

60  Barrels  of  oil,  32  gallons  each,  at  $1.08  per  gall.  

Amount,        .        .        .  $ 
Received  the  above  in  full, 


JACOB  THRIFTY. 


DENOMINATE    NUMBERS.  147 


DENOMINATE    NUMBERS. 

180.  A   DENOMINATE  NUMBER   is  a  denominate  unit,  or  a 
collection  of  such  units. 

181.  Two  numbers  are  of  the  same  denomination,  when 
.hey  have   the   same   unit ;    and   of  different   denominations, 
when  they  have  different  units. 

182.  A  COMPOUND  DENOMINATE  NUMBER   is  one  expressed 
by  two  or  more  different  units ;  as,  3  yards  2  feet  3  inches. 

183.  A  SCALE    is  a  connecting  link  between  two  denomi- 
nations.    Its  value,  is  the   number  of  units  of  the  less  de- 
nomination, which  make  1  unit  of  the  greater. 

184.    Kinds  of  Units. 

There  are  eight  different  Units  of  Arithmetic : 
I.   UNITS  OF  ABSTRACT  NUMBER  ; 
II.    UNITS  OF  CURRENCY  ; 
III.    UNITS  OF  LENGTH  ; 
IV'.    UNITS  OF  SURFACE  ; 
Y.   UNITS  OF  VOLUME,  OR  CAPACITY  ; 
VI.    UNITS  OF  WEIGHT  ; 
VII.    UNITS  OF  TIME  ; 
.     VIII.   UNITS  OF  CIRCULAR  MEASURE. 

I.    ABSTRACT  NUMBERS. 

185.    An   ABSTRACT   NUMBER    is    one   whose    unit    is    not 
named. 

Table. 

10  Units make  1  Ten. 

10  Tens 1  Hundred. 

10  Hundred 1  Thousand. 

10  Thousand 1  Ten-thousand.  , 

&c.,  &c. 


148 


DENOMINATE  NUMBERS. 


Thons. 
Ten-thous. 

1     =     10     = 


Table  Reversed. 

Ten.  Units. 

Hund.                      1  =  10. 

1       =            10  =  100. 

10     =       100  =  1000. 

100     =     1000  =  10000. 


SCALE. — 10,  and  uniform. 

II.    CURRENCY. 

I.     UNITED    STATES    CUKKENC.Y. 

186.   The  United  States  Currency,  is  the  Decimal  Currency 
established  by  a  law  of  Congress,  in  1792. 


Table. 

10  Mills  (m.)    .    .    .     make  1  Cent, 

10  Cents 1  Dime, 

10  Dimes 1  Dollar, 

10  Dollars 1  Eagle, 

Table  Reversed. 


marked  ct. 
.    .    .     d. 

.    .    .      $ 
E. 


d. 

ct 

1  = 

Bl. 

10. 

$ 

1  = 

10  = 

100. 

E. 

1 

=  10  «= 

100  = 

1000 

1  = 

10 

=  100  = 

1000  = 

10000 

SCALE. — 10,  and  uniform. 


180.  What  is  a  denominate  number? 

181.  When  are  two  numbers  of  the  same  denomination  ?    When 
of  different  denominations  ? 

182.  What  is  a  compound  denominate  number? 

183.  What  is  a  scale?    What  is  its  value? 

184.  How  many  kinds  of  units  are  there  in  Arithmetic?    Name 
them. 

185.  What  is  an   abstract   number?    Repeat  the  Table.    What 
are  the  scales,  in  Abstract  Numbers  ?    Are  they  uniform,  or  vari- 
able? 

186.  What  is  United  States  Currency?    What  are  its  denomina- 
'tions?    Repeat  the  Table.    What  are  the  scales? 


REDUCTION.  149 


II.     ENGLISH   CURRENCY. 

187.  The    English    Currency,   is    the    Currency   of   Great 
Britain. 

*  Table. 

4  Farthings  (far.}  .     make  1  Penny,    .    .    .  marked  d. 

12  Pence 1  Shilling, 8. 

20  Shillings 1  Pound,  or  sovereign,  .    £ 

21  Shillings 1  Guinea. 

Table  Reversed. 

d.  far. 

1    =         4. 

£  1   =     12  =     48. 

1   =  20  =  240  =  960. 

SCALES. — 1.  Beginning  at  the  lowest  unit,  the  scales  are  4,  12, 
and  20.  If  we  hegin  at  the  highest  unit,  the  order  is  reversed, 
and  the  scales  are  20,  12,  and  4.  The  connecting  link,  or  scale, 
between  any  two  denominations,  is,  h<5Wever,  the  same  in  both 
cases. 

2.  The  scales  are  uniform,  only  in  abstract  and  decimal  num- 
bers; in  all  others,  variable. 

3.  Farthings  are  generally  expressed  in  fractions  of  a  penny: 
Thus,    Ifar.  =  Jd. ;   2far.  =  ^d. ;    3far.  =  £d. 

4.  By  reading  the  second  table  from  left  to  right,  we  can  see 
the  value  of  any  unit  expressed  in  each  of  the  lower  denomina- 
tions.    Thus,    Id.  =  4far.;     Is.  =  12d.  =  48far.;     £1  =  20s.  = 
240d.  =  960far. 

Reduction. 

188.  REDUCTION    is  the  operation  of  changing  the  unit  of 
a  number,  without  altering  the  value  of  the  number. 

187.  What  is  English  Currency  ?    What  are  its  denominations  ? 
Repeat  the  Table.    What  are  the  scales?    Are  they  uniform,  or 
variable  ? 

188.  What  is  Reduction  ? 


150  DKNOMINATK    NUMBERS. 

189.  REDUCTION  DESCENDING    is  changing  the  unit  from  a 
greater  to  a  less. 

190.  REDUCTION  ASCENDING   is  changing  the  unit  from-  a 
less  to  a  greater. 

191.    Reduction  Descending. 

1.  Reduce  £27  6s.  8|d.,  to  the  denomination  of  farthings. 

ANALYSIS. — Since  there  are  20  shil- 
lings in  £1,  in  £27  there  are  27  times 

20    shillings,    or   540    shillings,    and    6          ^27     6s-    8d-    2far- 
shillings  added,  make  546s.     Since   12 
pence  make  1  shilling,  we  next  multi-  546s. 

ply  by  12,   and  then  add  8d.  to  the  12 

product,   giving    6560   pence.     Since  4 
farthings  make  1  penny,  we  next  mul- 
tiply by  4,  and  add  2  farthings  to  the 
product,  giving  26242  farthings  for  the         26242,  Ans. 
answer:     Hence,  the  following 

Rule. 

I.  Multiply  the  highest  denomination  by  the  scale,   and 
add  the  units,  if  any,  of  the  next  lower  denomination  : 

II.  Proceed  in  the  same  manner  through  all  the  denom- 
inations, till  the  number  is  brought  to  the  required  unit. 

192.    Reduction  Ascending. 

1.    In   26242   farthings,  how  many  pounds,   shillings,   and 

pence  ? 

OPERATION. 

ANALYSIS. — Since  4  farthings  make  1  A  \  25942 

penny,  we  first  divide  by  4.     Since  12  - — 

pence  make  1  shilling,  we  next  divide  12)_6560  .  2far.  rem 

by  12.    Since  20  shillings  make  1  pound,  2|0  )  54|6  .  8d.  rem. 
we   next   divide  by   20,  and  find  that 

26242   farthings  =  £27   6s.    8d.   2far. :  2>I  •  •   6s-  rem- 

Hence,  the  following  ^ns>  £27  6s.  8d.  2far. 


REDUCTION.  151 

Rule. 

I.  Divide  the  given  number  by  the  scale,  and  set  down 
the  remainder,  if  there  be  one : 

II.  Divide  the  quotient  by  the  next  scale,  and  set  aside 
the  remainder:  proceed  in  the  same  way,  through  all  the 
denominations;   and  the  last  quotient,  with  the  several  re- 
mainders annexed,  will  be  the  answer. 

PROOF. — The  proof,  in  either  case,  is  made  by  reversing 
the  operation. 

Examples. 

1.  Reduce  £15  7s.  6d.,  to  pence. 

OPERATION.  PROOF. 

£15     7s.     6d  12 )  3690 

2|0  )30|7  .  6d.  rem. 

jl  15  .  .  7s.  rem. 

—  Ans.  £15  1,.  6d. 

2.  In  £31  8s.  9d.  3far.,  how  many  farthings  ?     Proof. 

3.  In  £87  14s.  8|d.,  how  many  farthings  ?     Also  proof. 

4.  In  £407  19s.  lljd.,  how  many  farthings? 

5.  In  80  guineas,  how  many  pounds  ? 

6.  In  1549far.,  how  many  pounds,  shillings,  and  pence  ? 

7.  In  6169  pence,  how  many  pounds  ? 

in.    UNITS  OF  LENGTH. 
I.     LONG   MEASURE. 

193.   This  measure  is  used  to  measure  distances,  lengths, 
breadths,  heights,  depths,  &c. 

189.  What  is  .Reduction  Descending  ? 

190.  What  is  Reduction  Ascending? 

191.  What  is  the  rule  for  Reduction  Descending? 

192.  What  is  the  rule  for  Reduction  Ascending? 

193.  What  is  Long  Measure  used  for?    What  are  its  Units?    Re- 
peat the  Table.    What  are  the  Scales  ? 


DENOMINATE    NUMBERS. 


Table. 

s  (in.)      .     .      make    1  Foot,    ....    marked  ft. 
1   Yard  11  d. 

\,  or  16}  Feet,     .     . 
ngs,  or  320  Rods,    . 

1  Rod,     

.     .     rd. 

1  Furlong 

.   fur. 

1  Mile,    

.    .     mi. 

1  League         . 

.    .      L. 

te  Miles  (nearly),  .or  ) 
•aphical  Miles,     .     .) 

1  Degree  of  the  ) 
Equator,      ) 

.  deg.  or  ° 

12    Jnche 

3    Feet 

5}  Yards 
40  Rods 

8    Furlo 

3  Miles 
69}  Statin 
60  Geogi 
300  Degrees, a  Circumference  of  the  Earth. 

Table  Reversed. 


ft 

In. 

yd. 

1 

=    12. 

rd. 

1 

=    3 

=    36. 

far. 

1 

-    5J 

=   16J 

=   198. 

mi. 

1 

=   40 

=  220 

~  660 

=  7920. 

1 

=  8 

=  320 

=  1760 

=  5280 

=  63360. 

NOTES. — 1.    A  fathom  is  a  length  of  six  feet,  and  is  generally 
used  to  measure  the  depth  of  water.     A  pace  is  three  feet. 

2.  A  hand  is  4  inches,  used  to  measure  the  height  of  horses. 

3.  SOALKS. — The  scales,  beginning  at  the  smallest  unit,  are,  12, 
3,  5.},  40,  and  8. 

4.  The  geographical  mile  is  equal  to  a  minute  of  one  of  the 
great  circles  of  the  earth. 

Examples. 

1.    How   many   inches   in     |        2.    In   1365    inches,   how 

many  rods  ? 

OPERATION. 
12)  1365 

3)  113  feet  9  in. 

5J)-37  yds.  2  feet. 
11  )74 

6  rd.  8  half  yd. -4  yd. 
Ans.  6  rd.  4  yd.  2  ft.  9  in. 


6  rd.  4  yd.  2  ft.  9  in.? 

OPERATION. 

6rd.  4  yd.  2ft.  9  in. 


3 
34 

37  yards. 
3 

113  feet. 
12 


1365  inches. 


REDUCTION.  153 

3.  In  59  mi.  7  fur.  38  rd.,  how  many  feet  ? 

4.  In  115188  rods,  how  many  miles  ? 

5.  In  719  mi.  16  rd.  6  yd.,  how  many  feet  ? 

6.  In  118°,  how  many  miles? 

7.  In  54°  45  mi.  7  fur.  20  rd.  4  yd.  2  ft.  10  in.,  how 
many  inches  ? 

8.  In  481401716  inches,  how  many  degrees,  &c.  ? 

9.  If  a  river  is  65  fathoms  5*  feet  4  inches  deep  :    what 
is  its  depth  in  inches  ?   what  is  its  height  ? 

10.    If  a  horse  is  15|  hands  high,  how  much  in  feet  and 
inches  ? 


194.  The  Surveyors'  or  Guntcr's  Chain  is  generally  used  in 
surveying  land.  It  is  4  rods,  or  66  feet  in  length,  and  is 
.divided  into  100  links. 

Table. 

7.92  Inches  ....      make    1  Link,     .    .    .    marked    I. 

100  Links,  or  66  feet,  ...    1  Chain, c. 

80  Chains      ......    1  Mile, mi. 

Table  Reversed. 

1.  in. 

ft        1  *=    7-92. 

m.      1  =    66  =   100  =    792. 

1  z=  80  =  5280  =  8000  =  63360. 

SCALES. — The  Scales,  beginning  at  the  lowest  unit,  are  7.92, 
100,  and  80. 

Examples. 

1.  In  560  1.,  how  many  chains  ? 

2.  In  40  c.  65  1.,  how  many  feet  and  inches  ? 

3.  A  field,  regular  in  form,  is  65  c.  15  1.  long,  and  21  c.  14  1. 
broad  :  what  is  the  distance  around  it  ? 

194.  What  chain  is  used  in  land-surveying  ?  What  is  its  length  ? 
How  is  it  divided?  Repeat  the  Table.  What  are  the  Scales? 

7* 


154  DENOMINATE   NUMBERS. 


HI.     CLOTH   MEASURE. 

195.  CLOTH  MEASURE,  is  used  for  measuring  all  kinds  of 
cloth,  ribbons,  and  other  things  sold  by  the  yard. 

Table. 

2%  Inches  (in.)     .  * .      make  1  Nail,      .     .    .  marked  no,. 

4    Nails 1  Quarter  of  a  yard,     .    qr. 

3  Quarters 1  Ell  Flemish, .     .     .  E.  Fl. 

4  Quarters 1  Yard, yd. 

5  Quarters 1  Ell  English,  .     .     .    E.  E. 

6  Quarters 1  Ell  French,    .    .     .    E.  F. 

Table  Reversed. 

na.  in. 

qr.  1     -        2J. 

yd.  1=4=9. 

EFL  1   =  4  =  16  =  36. 

RE.       1   =  >   =  3  =  12  =  27. 

E.R       1   =   lf=  11=  5  =  20  =  45. 

1   =   lj=  2  =  lj=   6  =  24  =  54. 

SCALES. — 1.  The  scales,  beginning  at  the  least  unit,  and  then 
reckoning  from  the  quarter-yard,  are,  2|,  4,  4,  3,  5,  6. 

2.  The  yard  of  Cloth  Measure,  is  the  yard  of  Long  Measure, 
and  is  equal  to  36  inches. 

Examples. 

1.  In  35  yards  3  qr.  3  na.,  how  many  nails  ? 

2.  In  575  nails,  how  many  yards  ? 

3.  In  49  E.  E.,  how  many  nails  ? 

4.  In  51  E.  Fl.  2  qr.  3  na.,  how  many  nails  ? 

5.  In  3278  nails,  how  many  yards  ? 

6.  In  340  nails,  how  many  Ells  Flemish  ? 

7.  In  4311  inches,  how  many  E.  E.  ? 

195.  For  what  is  Cloth  Measure  used  ?  What  arc.  its  denomma 
tio'ns?  Repeat  the  Table.  What  are  the  units  of  this  measure? 


Square 
Foot 


REDUCTION.  155 

IV.    UNITS  OF  SURFACE. 

I.     SQUARE    MKA8UKE. 

196.  SQUARE  MEASURE,  is  used  in  measuring  surfaces, 
which  combine  length  and  breadth. 

The  unit  of  this  measure,  is  a  square,  constructed  on  the 
unit  of  length. 

A  square,  is  a  figure  bounded  by  four  * foot 

equal  sides,  at  right  angles  to  each  other. 
If  each  side  be  one  foot,  the  figure  is 
called,  a  square  foot. 

If  the  sides  of  the  square  be  each  one 

yard,  the  square  is  called,  a  square  yard.  J 

If  two  adjacent  sides  of  the  square  yard  °° 

be  divided  into  feet,  and  through  the  points  ^ .       . 

of  division,  lines  be  drawn  parallel  to  the  S. 

other  sides,  the  large  square  will  contain  **  \  y^  —  8  feet 
9   small    squares,  which   are   square   feet. 
Therefore,  the  square  yard  contains  9  square  feet. 

The  number  of  small  squares  that  is  contained  in  any  large 
square,  or  in  any  figure  whose  opposite  sides  are  parallel,  is 
always  equal  to  the  product  of  the  length  and  breadth.  As  in 
the  figure,  3x3  =  9  square  feet.  The  number  of  square  inches 
contained  in  a  square  foot,  is  equal  to  12  x  12  =  144. 

Table. 

144    Square  Inches  (sq.  in.}  make  1  Square  Foot,   .    marked  sq.  ft. 

9    Square  Feet 1  Square  Yard,  .     .     .     .    sq.  yd. 

30J  Square  Yards 1  Square  Rod,  or  Perch,     .     P. 

40    Square  Rods  or  Perches     .  1  Rood, R. 

4    Roods     ........  1  Acre, A. 

640    Acres 1  Square  Mile, M. 


196.  For  what  is  Square  Measure  used?  What  is  the  unit  of 
this  measure  ?  What  is  a  square  ?  If  each  side  be  one  foot,  what 
is  it  called?  If  each  side  be  a  yard,  what  is  it  called?  How  many 
square  feet  does  the  square  yard  contain  ?  How  is  the  number  of 
sinall  squares  contained  in  a  large  square  found  ?  Repeat  the  Table. 
What  are  the  scales? 


156  DENOMINATE   NUMBERS. 

Table  Reversed. 

sq.  ft.        sq.  in. 

6q.yd.        1  =       144. 

P.  1  =  9  =  1296. 

R     1  =  30J=  272J=  39204. 

^    1  =  40  =  1210  -  10890  =  1568160. 

1  =  4  =  160  =  4840  =  43560  =  6272640. 

SCALES. — The  scales,  beginning  at  the  lowest  unit,  are,  144,  9, 
80J,  40,  and  4. 

197.  Surveyors  estimate  the  area  of  land  in  Square  Miles, 
Acres,  Roods,  and  Perches. 

Table. 

16  Perches make  1  Square  Chain. 

40  Perches,  or  2£  Square  Chains   .    .  1  Rood. 

4  Roods 1  Acre. 

640  Acres 1  Square  Mile. 

Table  Reversed. 

sq.  oil.  P. 

p,               1   =  16. 

A.               1   =         2i=  40. 

8q.mi.          1   =         4   =       10  =  160. 

1  =  640  =  2560  -  6400  =  10240. 

SCALES. — The  scales,  beginning  at  the  lowest  unit,  are,  16,  2£, 
4,  and  640. 

Examples. 

1.    How   many  perches    in        2.  How  many  square  miles, 


32  M.  25  A.  3  R.  19  P.  ? 


OPERATION. 


32  M.  25  A.  3  R.  19  P. 
640 


640  )  20505     .  25  A. 


82023  roods. 
40 

3280939  perches. 


acres,  &c.,  in  3280939  P.? 

OPERATION. 

40  )  3280939  .  19  P. 
4  )  82023  .  3  R. 
[05 
32 


40  Ans.  32  M.  25  A.  3  R.  19  P. 


REDUCTION.  157 

3.  In  19  A.  2  R.  37  P.,  how  many  square  rods  ? 

4.  In  175  square  chains,  how  many  square  feet? 

5.  In  37456  square  inches,  how  many  square  feet  ? 

6.  In  14972  perches,  how  many  acres  ? 

7.  In  3674139  perches,  how  many  square  miles  ? 

8.  Mr.  Wilson's   farm  contains   104  A.  3  R.  and  19  P. ; 
he  paid  for  it  at  the  rate  of  75  cents  a  perch :  what  did  it 
cost? 

9.  The  four  walls  of  a  room  are  each   25  feet  in  length 
and  9  feet  in  height,  and  the  ceiling  is  25  feet  square :   how 
much  will  it  cost  to  plaster  it,  at  9  cents  a  square  yard  ? 

V.    UNITS  OF  VOLUME  OK  CAPACITY. 
I.     CUBIC    MEASURE. 

198.  CUBIC  MEASURE,  is  used  for  measuring  solids  ;  as 
stone,  timber,  earth,  and  other  things,  in  which  the  three 
dimensions  of  length,  breadth,  and  thickness,  are  considered. 

The  unit  of  this  measure  is  a  cube  whose  edge  is  the  unit 
of  length. 

A  cube  is  a  figure  bounded  by  six  equal  squares,  called 
faces;  the  sides  of  the  square  are  called  edges. 

A  cubic  foot  is  a  cube,  each  of  whose  faces  is  a  square 
foot ;  its  edges  are  each  1  foot. 

A  cubic  yard  is  a  cube,  each  of 
whose  edges  is  1  yard. 

The  base  of  a  cube  is  the  face  on 
which  it  stands.  If  the  edge  of  the 
cube  is  one  yard,  its  base  will  contain 
3x3  =  9  square  feet ;  therefore,  9  3  fL 

cubic  feet  can  be  placed  on  the  base, 
and  hence,  if  the  block  were  1  foot  thick,  it  would  contain  9 
cubic  feet ;  if  it  were  2  feet  thick,  it  would  contain  2  tiers 
of  cubes,  or  18  cubic  feet ;  if  it  were  3  feet  thick,  it  would 
contain  27  cubic  feet.  Applying  similar  reasoning  to  other 
like  solids,  we  conclude,  that, 


158 


DENOMINATE   NUMBERS. 


The  contents   of  a  body   are  found  by  multiplying  the 
length,  breadth,  and  thickness  together. 

Table. 

1728  Cubic  Inches  (cu.  in.)  make     1  Cubic  Foot,    .    marked  cu.  ft. 

27  Cubic  Feet 1  Cubic  Yard,    ....    cu.  yd. 

40  Feet  of  round,  or         )  1  „  T 

50  Feet  of  hewn  Timber,  f 

T. 


16  Cubic  Feet 

1  Cord  I 

foot,      .    . 

.  c. 

8  Cord  Feet,  < 

3r  [       ....     1  Cord, 

128  Cubic  Feet, 

Table  Reversed. 

Cu.  ft 

cu.  in. 

C.ft. 

-\    

1728. 

Cu.  yd.       1 

=     16  = 

27648. 

T.  rd.  T.       1          — 

27  = 

46656. 

T. 

hewn  T.        1          ==          2^ 

=.     40  = 

69120. 

T.  ship. 

1              =              31 

=     50  = 

86400. 

Cord.           1 

—                                        2f 

=     42  = 

72576. 

1 

=                         8 

=   128  = 

221184. 

NOTES. — 1.  A  cord  of  wood  is  a  pile  4  feet  wide,  4  feet  high, 
and  8  feet  long. 

2.  A  cord  foot  is  1  foot  in  length  of  the  pile  which  makes  a 
cord. 

3.  A  ton  of  round  timber,  when  square,  is  supposed  to  produce 
40  cubic  feet;  hence,  one-fifth  is  lost  by  squaring. 

Examples. 


1.  In  15  cu.  yd.  18  cu.  ft, 
16  cu.  in.,  how  many  cubic 
inches  ? 

OPERATION, 
cu.  yd.      cu.  ft.       cu.  in. 

15         18         16 

27 

113 
31 

423x1728  -1-16  =  730960. 


2.  In  730960  cubic  inch- 
es, how  many  cubic  yards, 
Ac.? 

OPERATION. 

1728  )  730960  cu.  in. 
27  )423  cu.  ft.  1 
15  cu.  yd.  18 

cu.  yd.       cu.  ft.       cu.  in. 

Ans.     15         18         16 


REDUCTION.  159 

3.  How  many  blocks,  of  1  cubic  inch,  can  be  sawed  from 
a  cube  of  7  feet,  if  there  is  no  waste  in  sawing  ? 

4.  In   25   cords    of  wood,   how   many   cord   feet  ?     How 
many  cubic  feet  ? 

5.  How  many  cords  of  wood  in  a  pile  28  feet  long,  4  feet 
wide,  and  6  feet  in  height  ? 

6.  In  174964  cord  feet,  how  many  cords? 

7.  In   17645900   cubic   inches,    how   many   tons   of  hewn 
timber  ? 

II.     LIQUID    MEASURE. 

199.  LIQUID  MEASURE,  is  used  for  measuring  all  liquids. 
Formerly  some  of  them  were  measured  by  Beer  Measure  ; 
but  that  measure  is  now  not  much  used. 


Table. 


4  Gills  (gi.) .     .     .      make 

2  Pints 

4  Quarts 

81 J  Gallons 

2  Barrels,  or  63  Gallons  . 

2  Hogsheads 

2  Pipes 


Pint,  ....  marked  pt. 

Quart, gt. 

Gallon, gal. 

Barrel,    .     .     .   bar.  or  bbl. 
Hogshead,  ....     hhd. 

Pipe, pi. 

Tun, tun. 


Table  Reversed 

pt  gi. 

qt,  1    =  4. 

gal.  I    =  2    =  8. 

bar.  1    =  4    =  8    =         32. 

hhd.  1  =  31J  =  126  =  252  =  1008. 

pi>    1  =  2  =  63  =  252  =  504  =  2016. 

tun.   1  -  2  =  4  =  126  =  504  =  1008  =  4032. 

1  =  2  =  4  =  8  =  252  =  1008  =  2016  =  8064. 

NOTE. — The  standard  unit,  or  gallon  of  Liquid  Measure,  in  the 
United  States,  contains  231  cubic  inches. 

198.  For  what  is  Cubic  Measure  used  ?  What  are  its  denomina- 
tions ?  What  is  a  cord  of  wood  ?  What  is  a  cord  foot  ?  What  Is 
a  cube  ?  What  is  a  cubic  foot  ?  What  is  a  cubic  yard  ?  How  many 
cubic  feet  is  a  cubic  yard  ?  What  are  the  contents  of  a  solid  equal 
to?  Repeat  the  Table.  What  are  the  Scales? 


160 


DENOMINATE   NUMBERS. 


Examples. 

1.    In  5  tuns   3  hogsheads         2.    In    1466    gallons,    how 
1 7  gallons  of  wine,  how  many  j  many  tuns,  &c.  ? 
gallons  ? 

OPERATION. 

63)  1466 

17  gal. 
3hhd. 


OPERATION. 

5  tuns  3  hhd.  17  gal. 
4 


23 
63 


76 
139 


4)23 
5 


Ans.  5  tuns  3  hhd.  17  gal. 
1466  gallons. 

3.  In  12  pipes,    1   hogshead,   and   1  quart   of  wine,  how 
many  pints  ? 

4.  In  10584  quarts  of  wine,  how  many  tuns  ? 

5.  In  201632  gills,  how  many  tuns? 

6.  What  will  be  the  cost  of  3  hogsheads,  1  barrel,  8  gal- 
lons, and  2  quarts  of  vinegar,  at  4  cents  a  quart  ? 


III.     DRY   MEASURE. 

200.   DRY  MEASURE,  is  used  in  measuring  all  dry  articles, 
such  as  grain,  fruit,  salt,  coal,  &c. 

Table. 

*  2  Pints  ( pt.)  .    .    .      make  1  Quart,    .    .    .   marked  qt. 

8  Quarts    .......  1  Peck, pk. 

4  Pecks 1  Bushel, bu. 

36  Bushels 1  Chaldron, ch. 

Table  Reversed. 

qt  pt. 

pk.  1     -  2. 

bn.  1   =          8   =        16. 

ch.  1   =       4  =       32  =       64. 

1  =  36  =  144  =r  1152  =  2304. 

SCALES. — The  scales,  beginning  with  the  lowest  unit,  are,  2, 
8,  4,  and  36. 


REDUCTION. 


161 


NOTES. — 1.  The  standard  bushel  of  the  United  States,  is  the 
Winchester  bushel  of  England.  It  is  a  circular  measure,  18^ 
inches  in  diameter  and  8  inches  deep,  and  contains  2150^  cubic 
inches,  nearly. 

2.   A  gallon,  Dry  Measure,  contains  268|  cubic  inches. 


1.  How  many  quarts  are 
there  in  65  ch.  20  bu.  3  pk. 
7qt.? 

OPERATION. 

65  ch.  20  bu.  3  pk.  7  qt. 
36 

390 
197 

2360 
4 

9443 

8 


Examples. 

2.    How    many   chaldrons, 
&c.,  in  75551  quarts.  ? 


OPERATION. 

8)  75551 

4  )  9443 

36  )  2360 

65 


7  qt. 

3pk. 

20  bu. 


Ans.  65  ch.  20  bu.  3  pk.  7  qt. 


75551  quarts. 

3.  In  372  bushels,  how  many  pints  ? 

4.  In  5  chaldrons  31  bushels,  how  many  pecks  ? 

5.  In  17408  pints,  how  many  bushels  ?- 

6.  In  4220  pints,  how  many  chaldrons  ? 


VI.     UNITS    OF    WEIGHT. 
I.    AVOIKDUPOIS  WEIGHT. 

201.   By  this  weight  all  articles  are  weighed,  except  gold, 
silver,  jewels,  and  liquor. 

199.  What  is  measured  by  Liquid  Measure  ?    What  are  its  de- 
nominations ?    Repeat  the  Table.    What  are  the  units  of  the  scale  ? 
What  is  the  standard  wine  gallon  ? 

200.  What  articles   are   measured  by  Dry  Measure  ?    What  are 
its    denominations  ?     Repeat    the    Table.    What    are    the    scales  ? 
What  is  the  standard  bushel  ?    What  are  the  contents  of  a  gallon  ? 


DENOMINATE    N  UMBKRS. 


Table. 

16  Drams  (dr.)    .     .      make  1  Ounce,   .     .     .   marked  oz. 

16  Ounces 1  Pound, Ib. 

25  Pounds 1  Quarter, qr. 

4  Qxiarters 1  Hundredweight,      .     .  cwt. 

20  Hundredweight  ....  1  Ton, T. 

Table  Reversed. 


Ib. 

oz. 

1  = 

dr. 

16. 

qr. 

1  

16  = 

256. 

wrt 

1  = 

25  = 

400  = 

6400. 

1  = 

4  — 

100  = 

1600  = 

25600. 

T. 

1   =   20  =  80  =  2000  =  32000  =  512000. 

SCALES. — The  scales,  beginning  with  the  least  unit,  are,  16,  16, 
25,  4,  and  20. 

NOTES. — 1.  The  standard  Avoirdupois  pound  is  the  weight  of 
27.7015  cubic  inches  of  distilled  water. 

2.  By  the  old  method  of  weighing,  adopted  from  the  English 
system,    112   pounds  were   reckoned  for   a   hundredweight;   but 
now,  the  laws  of  most  of  the  States,  as  well  as  general  usage, 
fix  the  hundredweight  at  100  pounds. 

3.  A  ton  of  coal  at  the  mines,  is  reckoned  at  2240  Ib.,  but  at 
the  yards,  at  2000  Ib. 

Examples. 

1.  How  many  pounds  are 
there  in  15  T.  8  cwt.  3  qr. 
15  Ib.  ? 

OPERATION. 


15  T.  8  cwt.  3  qr.  15  Ib. 
20 

308  cwt. 
4 


1235  qr. 
25 


6180 
2471 

30890  Ib. 


51b.  added. 
1  ten  added. 


2.    In  30890  pounds,  how 
many  tons  ? 


OPERATION. 
25  )  30890 

4  )  1235  qr. 
20  )  308  cwt. 
15  T. 


15  Ib. 
3  qr. 

8  cwt. 


Ans.  15  T.  8  cjrt.  3  qr.  15  Ib. 


REDUCTION. 

3.  In  5  T.  8  cwt.  3  qr.  24  Ib.  13  oz.  14  dr.,  how  many  drams  ? 

4.  In  28  T.  4  cwt.  1  qr.  21  Ib.,  how  many  ounces? 

5.  In  2790366  drams,  how  many  tons? 

6.  In  903136  ounces,  how  many  tons  ? 

7.  In  3124446  drains,  how  many  tons  ? 

8.  In  93  T.  13  cwt.  3  qr.  8  Ib.,  how  many  ounces  ? 

9.  In  108910592  drams,  how  many  tons  ? 

10.  What  will  be  the  cost  of  11  T.  17  cwt.  3  qr.  24  Ib.  or 
hay,  at  half  a  cent  a  pound?     How  much  would  that  be  a 
ton? 

11.  What  is  the  cost  of  2  T.  13  cwt.  3  qr.  21  Ib.  of  beef, 
at  8  cents  a  pound  ?     How  much  would  that  be  a  ton  ? 

II.     TKOY   WEIGHT. 

202.  Gold,  silver,  jewels,  and  liquors  are  weighed  by  Troy 
Weight. 

Table. 

24  Grains  (gr.)    .    .      make     1  Pennyweight,  marked  pwt. 

20  Pennyweights     ....     1  Ounce, oz. 

12  Ounces 1  Pound, Ib. 

Table  Reversed. 

pwt  gr. 

1   =       24. 

lb.  1   =     20   =     480. 

1   -   12  =  240  =  5760.  . 

SCALES. — The  scales,  beginning  with  the  lowest  unit,  are,  24,  20, 
and  12. 

NOTE.— The  standard  Troy  pound,  is  the  weight  of  22.794377 
cubic  inches  of  distilled  water.  It  is  less  than  the  pound  Avoir- 
dupois. 

201.  For  what  is  Avoirdupois  Weight  used?  How  is  the  Table 
to  be  read?  How  can  you  determine,  from  the  second  Table,  the 
value  of  any  unit  in  units  of  the  lower  denominations  ?  Name  the 
scales. 

NOTES. — 1.  What  is  fche  standard  Avoirdupois  pound? 

2.  What  is  a  hundredweight  by  the  English  method?  What  is 
a  hundredweight  by  the  United  States  method  ? 


164 


DENOMINATE   NUMBERS. 


Examples. 


1.  How  many  grains  are 
there  in  16  Ib.  11  oz.  15pwt. 
17  gr.  ? 

OPERATION. 

16  Ib.  11  oz.  15pwt.  17  gr. 
12 

203  ounces. 
20 


4075  pennyweights. 
24 


97817  grains. 


2.    In   97817   grains,  how 
many  pounds  ? 

OPERATION. 
24)  97817 

20  )  4075  pwt.     17  gr. 
12  )  203  oz.        15  pwt. 
16  Ib.        11  oz. 

Ans.  16  Ib.  11  oz.  15  pwt.  17  gr. 


3.  In  25  Ib.  9  oz.  20  gr.,  how  many  grains  ? 

4.  In  6490  grains,  how  many  pounds  ? 

5.  In  148340  grains,  how  many  pounds  ? 

6.  In  117  Ib.  9  oz.  15  pwt.  18  gr.,  how  many  grains? 

7.  In  8794  pwt.,  how  many  pounds  ? 

8.  In  6  Ib.  9  oz.  21  gr.,  how  many  grains  ? 

9.  In  1  Ib.  1  oz.  10  pwt.   16  gr.,  how  many  grains  ? 

10.  A  jewel  weighing  2  oz.  14  pwt.  18  gr.,  is  sold  for  half 
a  dollar  a  grain :  what  is  its  value  ? 


III.     APOTHECARIES     WEIGHT. 


203.  This  weight  is  used  by  apothecaries  and  physicians 
in  mixing  their  medicines.  But  medicines  are  generally  sold, 
in  the  quantity,  by  avoirdupois  weight. 


Table. 


marked  9. 

3. 


20  Grains  (gr.)    .    .      make  1  Scruple, 

3  Scruples •  1  Dram,    . 

8  Drams 1  Ounce, §. 

12  Ounces 1  Poiind, Ib. 


202.  What  articles  are  weighed  by  Troy  WeighJ  ?  What  are  its 
denominations  ?  Repeat  the  Table.  What  are  the  scales  ?  What 
is  the  standard  Troy  pound  ? 


REDUCTION. 


165 


Table  Reversed, 


3 

3 
1 

•  gr- 

=         20 

I 

1 

=           3 

=         60 

fij 

1 

=         8 

=         24 

=       480 

1     = 

12 

=       96 

=       288 

=     5760 

SCALES. — The  scales,  beginning  with  the  lowest  unit,   are  30, 
3,  8,  and  12. 

NOTE. — The  pound  and  ounce  are  the  same  as  the  pound  and 
ounce  in  Troy  weight. 


Examples 

1.    How   many   grains    in 
9ft  8§  63  23  12 gr.? 

OPERATION. 


9ft  8 1  63  23  12  gr. 
12 

116  ounces. 


934  drams. 
3 


2804  scruples. 
20 


2.    In  56092  grains,  how 
many  pounds  ? 

OPERATION. 

20  )  56092 

3)28049 

8  )  934  5 


9ft 


12  gr. 
23 
63 

85 


Ans.  9ft  8§  63  23  12  gr. 


56092  grains. 

3.  In  27  ft  9  5  6  3  1  3,  how  many  scruples  ? 

4.  In  94  ft  11  I  13,  how  many  drams  ? 

5.  In  8011  scruples,  how  many  pounds? 

6.  In  9113  drams,  how  many  pounds  ? 

7.  How  many  grains  in  12  ft  9  I  73  23  18  gr.  ? 

8.  In  73918  grains,  how  many  pounds? 


VII.    UNITS   OF  TIME. 

204.    TIME  is  a  part  of  duration.    The  time  in  which  the 
earth  revolves  on  its  axis  is  called  a  day*    The  tune  in  which 


Iflfl 


DENOMINATE    NUMBERS. 


it  goes  round  the  sun  is  365  days  and  6  hours,  nearly,  and 
is  called  ,a  solar  year. 

Time  is  divided  into  parts  according  to  the  following, 

Table. 


60  Seconds  (see.)  .     . 
60  Minutes  .... 

make     1  Miniate,     .    . 
...     1  Hour 

marked  m. 
hr 

24  Hours     .         .    . 

1  Day 

da 

7  Days 

1  Week 

wk 

52  Weeks  (nearly)  . 
365  Days  .... 

...     1  Year,     .     .     . 
1  Common  Year 

-    -    .   yr. 

fff- 

366  Days  . 

1  Leap  Year 

1IT 

12  Calendar  Months 
100  Years 

...     1  Year,     .     .     . 
1  Century.    , 

.    •    .   yr. 

a 

Table  Reversed. 


hr. 

1 

— 

60. 

de. 

1 

— 

60 

=2 

360. 

wk. 

1  = 

24 

rr 

1440 

— 

86400. 

1 

=  7  = 

168 

— 

10080 



604800. 

m. 

|  365  = 

8760 

— 

525600 

— 

31536000. 

12 

1  366  = 

8784 

— 

527040 

— 

31622400. 

SCALES. — The  scales,  beginning  with  the  lowest  unit,  are  60,  60, 
24,  7,  52,  and  12. 


WINTER, 


SPRING, 


SUMMER, 


AUTUMN, 


WINTER, 


Calendar  Year. 

1st  Month,  January,          has 


2d 
3d 
4th 
5th 

{6th 
7th 
8th 
(  9th 

hath 

(nth 

12th 


February, 

March, 

April, 

May, 

June, 

July, 

August, 

September, 

October, 

November, 

December, 


31  days. 

28  or  29  days. 

31  days. 

30  days. 

31  days. 

30  days. 

31  days. 
31  days. 

30  days. 

31  days. 

30  days. 

31  days. 

365  days  in  a  year. 


REDUCTION". 


167 


NOTES. — 1.  The  years  are  numbered  from  the  beginning  of  the 
Christian  Era.  The  year  is  divided  into  12  calendar  months, 
numbered  from  January:  the  days  are  numbered  from  the  begin- 
ning of  the  month :  hours,  frtfrn  12  at  night  and  12  at  noon. 

2.  The  length  of  the  solar  year,  is  365  da.  5  hr.  48  m.  48  sec., 
nearly ;  but  it  is  reckoned  at  365  days  6  hours. 

3.  Since  the  length  of  the  year  is  computed  at  365  days  and 
6  hours,  the  odd  6  hours,  by  accumulating  for  4  years,  make  1 
day,  so  that  every  fourth  year  contains  366  days.     This  is  called, 
Bissextile  or  Leap  Year.     The  leap  years  are  exactly  divisible  by 
4:    1864,  1868,  1872,  1876,  will  be  leap  years. 

4.  The  additional  day,  when  it  occurs,  is  added  to  the  month 
of  February,  so  that  this  month  has  29  days  in  the  leap  year. 

Thirty  days  hath  September, 
April,  June,  and  November; 
All  the  rest  have  thirty-one, 
Excepting  February,  twenty-eight  alone. 

Examples. 


1.    How  many  seconds  in 
365  da.  6  hr.  ? 

OPERATION. 

365  da.  6  hr. 
24 


2.    How  many  days,   &c., 
in  31557600  seconds  ? 

OPERATION. 

60)  31557600 

60  )  525960 

24  )  8766 

365         6  hr. 

Ans.  365  da.  6  hr. 


1466 
730 

8766 
60 

525960x60  =  31557600  sec. 

3.  If  the  length  of  the  year  were  365  da.  23  hr.  57  m. 
39  sec.,  how  many  seconds*  would  there  be  in  12  years  ? 

204.  What  are  the  denominations  of  Time?  How  long  is  a 
year  ?  How  many  days  in  a  common  year  ?  How  many  days  in  a 
leap  year?  How  many  calendar  months  in  a  year?  Name  them, 
and  the  number  of  days  in  each.  How  many  days  has  February 
in  the  leap  year?  How  do  you  remember  which  of  the  months 
hsve  80  days,  and  which  81? 


168  DENOMINATE   NUMBERS. 

4.  In  126230400  seconds,  how  many  common  years? 

5.  In  756952018  seconds,  how  many  common  years  ? 

6.  In  285290205  seconds,  how  many  years  of  365  da.  6  hr. 
each? 

7.  How  many  hours   in   any  year  from   the  31st  day  of 
March   to    the    1st   day   of  January   folio  whig,    neither   day 
named  being  counted  ? 

VIII.    CIRCULAR  MEASURE. 

205.  CIRCULAR  MEASURE,  is  used  in  estimating  latitude  and 
longitude,  and  also  in  measuring  the  motions  of  the  heavenly 
bodies. 

The  circumference  of  every  circle  is  supposed  to  be  divided 
into  360  equal  parts,  called  degrees.  Each  degree  is  divided 
into  60  minutes,  and  each  minute  into  60  seconds. 

Table 

60  Seconds  (")   .     .      make  1  Minute,   .    .     .    marked  '. 

60  Minutes 1  Degree, °. 

15  Degrees 1  Hour  Angle,    .    .    hr.  an. 

30  Degrees 1  Sign,  .......    «. 

12  Signs,  or  360  Degrees     .  1  Circle, e. 

Table  Reversed. 

1   =  60. 

'    far..,,.  1    =  60    =  3600.    ' 

B.  1  -  15  =  900  =  54000. 
e.  1  =  2  =  30  =  1800  =  108000. 
1  =  12  =  24  -  360  =  21600  =  1296000. 

Examples. 

1.  In  5s.  29°  25',  how  man£  minutes? 

2.  In  2  circles,  how  many  seconds  ? 

3.  In  27894  seconds,  how  many  degrees  ? 

4.  In  32295  minutes,  how  many  circles  ? 

5.  In  3  circles  16°  20',  how  many  seconds? 

6.  In  8  s.  16°  25",  how  many  seconds? 
t.  In  8589  seconds,  how  many  degrees  ? 


REDUCTION.  169 


Miscellaneous  Tables. 

COUNTING. 

12  Units,  or  things,     ....      make  1  Dozen. 

12  Dozen 1  Gross. 

12  Gross 1  Great  Gross. 

20  Things 1  Score. 

LENGTH. 

18  Inches 1  Cubit. 

22  Inches,  nearly, 1  Sacred  Cubit. 

WEIGHT. 

100  Pounds 1  Quintal  of  fish. 

196  Pounds 1  Barrel  of  flour. 

200  Pounds 1  Barrel  of  pork. 

14  Pounds  of  iron  or  lead 1  Stone. 

21£  Stones 1  Pig. 

8  Pigs 1  Fother. 

PAPER. 

24  Sheets 1  Quire. 

20  Quires 1  Ream. 

2  Reams 1  Bundle. 

2  Bundles 1  Bale. 

BOOKS. 
i 

The  terms,  folio,  quarto,  octavo,  duodecimo,  &c.,  indicate 
the  number  of  leaves  into  which  a  sheet  of  paper  is  folded. 

A  sheet  folded  in    2  leaves,  is  called,  a  folio. 

A  sheet  folded  in    4  leaves,  ...  a  quarto,  or  4to. 

A  sheet  folded  in    8  leaves,  ...  an  octavo,  or  8vo. 

A  sheet  folded  in  12  leaves,  ...  a  12mo. 

A  sheet  folded  in  16  leaves,  ...  a  16mo. 

A  sheet  folded  in  18  leaves,  .     .     .  an  18mo. 

A  sheet  folded  in  24  leaves,  ...  a  24mo. 

A  sheet  folded  in  32  leaves,  ...  a  32mo. 

205.  For  what  is  Circular  Measure  used?     Hew  is  every  circle 
supposed  to  be  divided  ?    Repeat  the  Table. 

8 


170  DENOMINATE   .NUMBERS. 


Miscellaneous  Examples. 

1.  How  many  square  inches   are   there  in  a  floor  that  is 
24  feet  long  and  18  feet  wide  ? 

2.  A  ceiling  is  24  ft.  long,  and  18  ft.  wide :   what  would 
be  the  cost  of  plastering  it,  at  50  cents  a  square  yard  ? 

3.  How  many  solid  feet  in  a  pile  of  wood  28  feet  long, 
4  feet  wide,  and  6  feet  in  height  ?     How  many  cords  ? 

4.  A  block  of  marble  is  9  feet  long,  2J  feet  wide,  1J  ft. 
thick :  how  many  cubic  feet  does  it  contain  ? 

5.  A  room  is   16  feet  long,   11  feet  wide,  and   10  feet 
high :  how  many  gallons  of  air  does  it  contain  ? 

6.  What  is  the  cost  of  a  pile   of  wood  that  is   82  feet 
long,  18  feet  high,  and  4  feet  thick,  at  $3.75  per  cord? 

7.  A  pile  of  bricks,   solid,  is   14   feet  long,  8  fee*  wide, 
and  12  feet  high:  how  many  bricks  are  there  in  the  pile,  a 
brick  being  8  inches  long,  4  inches  wide,  and  2  inches  thick  ? 

8.  How  many  bottles,  holding  1J  pints,  would  be  required 
to  contain  a  hogshead  of  wine  ? 

9.  How  many  pails   of  water,  each  holding  3  gallons   2 
quarts,  can  be  drawn  out  of  a  cistern  containing  12  hogs- 
heads 24  gallons  2  quarts  ? 

10.  In  372  j  bushels,  how  many  pints  ? 

11.  A  grocer  bought  2  bushels  of  peanuts  at  $3  a  bushel, 
and  sold  them  at  4  cents  a  pint :  ,did  he  make  or  lose,  and 
how  much  ? 

12.  In  4044896  square  inches  of  land,  how  many  acres, 
and  what  is  its  value,  a,t  $75.18}  per  acre  ? 

13.  How  many  acres  are  there  in  250  city  lots  of  ground, 
each  of  which  is  25  feet  by  100  ? 

14.  In  6  years  (of  5'2j  weeks  each),  32  wk.  6  da.  17  hr., 
how  many  hours  ? 

15.  In  811480",  how  many  signs  ? 

16.  In  2654208  cubic  inches,  how  many  cords? 

17.  In  18  tons  of  round  timber,  how  many  cubic  inches? 

18.  In  84  chaldrons  oi  coal,  how  many  bags  containing 
3£  pecks  ? 


MISCELLANEOUS    EXAMPLES.  171 

19.  In  302  ells  English,  how  many  yards  ? 

20.  In  24  hhd.  18  gal.  2  qt.  of  molasses,  how  many  gills? 

21.  In  76  A.  1  R.  8  P.,  how  many  square  inches  ? 

22.  In  445577  feet,  how  many  miles? 

23.  In  37444325  square  inches,  how  many  acres? 

24.  How  many  times  will  a  wheel,  16  feet  and  6  inches  in 
circumference,  turn  round  in  a  distance  of  84  miles?' 

25.  What  will  28  rods  129  square  feet  of  land  cost,  at 
$15  a  square  foot  ? 

26.  What  will  be  the  cost  of  a  pile  of  wood,  36  feet  long, 
6  feet  high,  and  4  feet  wide,  at  50  cents  a  cord  foot  ? 

27.  A  man  has  a  journey  to  perform  of  288  miles.     He 
travels  the  distance  in  12  days,  traveling  6  hours  each  day : 
at  what  rate  does  he  travel  per  hour? 

28.  How  many  yards  of  carpeting,  1  yard  wide,  will  carpet 
a  room  18  feet  by  20  ? 

29.  If  the  number  of  inhabitants  in  the  United  States  were 
24  millions,  how  long  will  it  take  a  person  to  count  them, 
counting  at  the  rate  of  100  a  minute  ? 

30.  A  merchant  wishes  to  bottle  a  cask  of  wine  contain- 
ing 126   gallons,   in  bottles  containing   1  j   pint   each :    how 
many  bottles  are  necessary? 

31.  There  is  a  cube,  or  square  piece  of  wood,  4  feet  each 
way :   how  many  small  cubes,  of  1  inch    each   way,   can  be 
sawed  from  it,  allowing  no  waste  in  sawing  ? 

32.  A  merchant  wishes   to  ship  285  bushels   of  flax-seed 
in  casks  containing  7  bushels  2  pecks  each  :    what  number  of 
casks  are  required  ? 

33.  The  square  measure  of  a  floor  is  648  sq.  ft.,  and  its 
width  is  18  ft.:   what  is  the  length? 

34.  The  square  measure  of  a  board  is  9  sq.  feet,  and  its 
length  is  12  feet :  what  is  its  width  ? 

35.  A  cellar,  34  feet  by  25  feet,  and  8  feet  deep,  is  to  be 
dug.     Supposing  that  3  cart-loads  of  earth  make  1  solid  yard, 
how  many  loads  of  earth  would  be  carted  ? 

36.  How  many  bricks  will  pave  a  sidewalk  25  ft.  by  10, 
the  dimensions  of  a  brick  being  8  in.,  4  in.,  and  2  in.  ? 


172  DENOMINATE    FRACTIONS. 


DENOMINATE   FRACTIONS. 

206.  A  DENOMINATE  FRACTION  is  one  in  which  the  unit  of 
the  fraction  is  a  denominate  number.  Thus,  |-  of  a  yard  is 
a  -denominate  fraction. 

207.*  REDUCTION  of  denominate  fractions  is  the  operation 
of  changing  a  fraction  from  one  unit  to  another,  without 
altering  its  value. 

CASE    I. 
208.    To  change  from  a  greater  unit  to  a  less. 

I.  In  f-  of  a  yard,  how  many  inches? 

ANALYSIS.— Since  yards  are  reduced  OPERATION. 

to   feet  by  multiplying  by  3,  and  feet       _  4 

are   reduced  to    inches  by  multiplying       _  x  $  X  X$  =  20  in. 
by  12,  if  £  be  multiplied  by  3  and  12, 
the  product  will  be  inches :    Hence, 

Rule. — Multiply  the  fraction  by  the  scales  till  you  reach 
the  required  unit. 

Examples. 

2.  Reduce  T7^  of  a  £  to  the  fraction  of  a  penny. 

3.  Reduce  -^-Q  of  a  £  to  the  fraction  of  a  farthing. 

4.  Reduce  73g-  of  an  Ell  Eng.  to  the  fraction  of  a  nail. 

5.  Reduce  g-fg-  of  a  hogshead  to  the  fraction  of  a  quart. 

6.  Reduce  ^-|Q-  of  a  bushel  to  the  fraction  of  a  pint. 

7.  Reduce  -^^  of  a  pound  Troy  to  the -fraction  of  a 
grain. 

8.  Reduce  2  ^  Q  of  a  cwt.  to  the  fraction  of  an  ounce. 

9.  Reduce  .125  cwt.  to  the  decimal  of  an  ounce. 

10.   Reduce  .3125  of  a  mile  to  the  decimal  of  an  inch. 

II.  Change  .1875  of  a  day  to  minutes. 

12.  Reduce  .29763  of  a  degree  to  the  decimal  of  a  second. 

13.  Reduce  .1723  Ib.  to  the  decimal  of  a  grain. 

14.  Reduce  2.333  £  to  pence. 


REDUCTION. 

CASE    II. 
209.    To  change  from  a  less  unit  to  a  greater. 

1.  In  20  inches,  bow  many  yards? 

ANALYSIS. — Inches  are  reduced  to  OPERATION. 
yards,  by  dividing  by  12  and  3  in 

succession.     To  divide  by  12  and  3,  20  x  J_  x  \  _  &  ^ 

is  the  same  as  multiplying  by  y^  and  1        '%.%       3 

£;  therefore,  2T°  x  TV  x  \  =  f  yd.  3 

Rule. — Divide  the  fraction  by  the  scales,  in  succession, 
till  the  required  unit  is  reached. 

Examples. 

2.  Reduce  £  of  a  gallon  to  the  fraction  of  a  hogshead. 

3.  Reduce  f  of  a  nail  to  the  fraction  of  a  yard. 

4.  Reduce  i  of  -J  of  a  foot  to  the  fraction  of  a  mile. 

5.  Reduce  f  of  3J  pwt.  to  the  fraction  of  a  pound  Troy. 

6.  Reduce  .3125  pt.  to  the  decimal  of  a  gallon. 

NOTE. — Divide  by  the  units  of  the  scale,  and  observe  the  rule 
for  pointing  in  division  of  decimals. 

7.  Change  .89725  oz.  to  the  fraction  of  a  cwt. 

8.  Change  .9825  of  a  penny  to  the  decimal  of  a  pound. 

9.  Reduce  6.875  seconds  to  the  decimal  of  a  day. 

10.  What  decimal  of  a  yard  are  27.9175  nails? 

11.  To  what  decimal  of  a  mile  are  262.318125  feet  equal? 

12.  Reduce  .009375  pt.  to  the  decimal  of  a  bushel  ? 

13.  Reduce  7  drams  to  the  decimal  of  a  Ib.  avoirdupois. 

14.  Reduce  .056  of  a  pole  to  the  decimal  of  an  acre. 

15.  Reduce  14  minutes  to  the  decimal  of  a  day. 

16.  Reduce  21  pints  to  the  decimal  of  a  peck. 

17.  Reduce  375678  feet  to  the  decimal  of  a  mile. 

18.  Reduce  .5  quarts  to  the  decimal  of  a  barrel. 

206.  What  is  a  denominate  number? 

207.  What  is  Reduction  of  Denominate  Fractions  ? 

208.  What  is  Case  I.  ?    Give  the  rule. 

209.  How  do  you  change  from  a  less  unit  to  a  greater? 


174: 


DENOMINATE   FRACTIONS. 


CASE     III. 

210.    To  find  the  value  of  a  common  fraction  in  integers 
of  lower  denominations. 

1.  What  is  the  value  of  f  of  a  pound  -Troy? 

OPERATION. 


ANALYSIS. — f  of  a  pound,  re- 
duced to  the  fraction  of  an 
ounce,  is  $  x  12  =  V  of  an 
ounce,  which  is  equal  to  9§ 
ounces :  f  of  an  ounce,  reduced 
to  the  fraction  of  a  pennyweight, 
20  =  6j°  of  a  pwt.,  or 


Numer. 


Denom.  5 


4 
12 

oz. 

48  (9 
45 

3 
20 


pwt 

12,  Ans. 


12  pwt.  5  )  6o 

60 
Rule. 

Multiply  the  numerator  of  the  fraction  by  the  scale,  and 
divide  the  product  by  the  denominator ;  if  there  is  a  re- 
mainder, treat  it  in  the  same  way,  till  the  required  denom- 
ination is  reached.  The  quotients  of  the  several  operations 
will  form  the  ansiver. 

Examples. 

1.  What  is  the  value  of  f  of  a  tun  of  wine  ? 

2.  What  is  the  value  of  T%  of  a  yard  ? 

3.  What  is  the  value  of  |   of  a  month  ? 

4.  What  is  the  value  of  f  of  a  chaldron  ? 

5.  What  is  the  value  of  J  of  a  mile  ? 

6.  What  is  the  value  of  •£$  of  a  ton  ? 

7.  What  is  the  value  of  -|  of  3  days  ? 

8.  What  is  the  value  of  i  of  £  of  6|  bushels  of  grain? 

9.  What  is  the  value  of  f  hhd.  ? 

10.  What  is  the  value  of  T7j  of  a  cwt.  ? 

11.  What  is  the  value  of  ^  of  a  hogshead  of  wine  ? 

12.  What  is  the  value  of  T73  of  an  acre  of  land  ? 

13.  What  is  the  value  of  }|  tons  ? 

14.  What  is  the  value  of  j-|  of  a  common  year? 


REDUCTION.  175 


CASE     IV. 

211.    To  find  the  value  of  a  decimal  in  integers  of  lower 
denominations. 

1.   Find  the  value  of  .890625  of  a  bushel. 

OPERATION. 

ANALYSIS. — Multiplying  the  decimal  by  4  890625  b 

(since  4  pecks  make  a  bushel),  we  have  3   , 

3.5625  pecks.    Multiplying  the  new  decimal 
by  8   (since   8   quarts  make   a  peck),  we  3.562500  pk. 

have    4.5    quarts.     Then,    multiplying   this  8 

last  decimal  by  2   (since  2  pints  make  a 
.quart),  we  have  1  pint. 

Ans.  3  pk.  4  qt.  1  pt.  1.000000  pt. 

Rule. 

I.  Multiply  the  decimal  by  the  scale,  pointing  off  as  in 
multiplication  of  decimal  fractions : 

II.  Multiply  the  decimal  part  of  the  product  as  before, 
and  continue   the   operations   to   the  lowest   denomination: 
the  integers  at  the  left,  form  the  answer. 

Examples. 

1.  What  is  the  value  of  .002084  Ib.  Troy  ? 

2.  What  is  the  value  of  .625  of  a  cwt.  ? 

3.  What  is  the  value  of  .625  of  a  gallon  ? 

4.  What  is  the  value  of  £.3375  ? 

5.  What  is  the  value  of  .3375  of  a  ton  ? 

6.  What  is  the  value  of  .05  of  an  acre  ? 

7.  What  is  the  value  of  .875  of  a  pipe  of  wine  ? 

8.  What  is  the  value  of  .125  of  a  hogshead  of  beer? 

9.  What  is  the  value  of  .375  of  a  year  of  365  days  ? 

210.  How  do  you  find  the  value  of  a  common  fraction  in  integers 
of  lower  denominations  ? 

211.  How  do  you  find  the  value  of  a  decimal  in  integers  of  lower 
denominations  ? 


176  DENOMINATE    FRACTIONS. 

10.  What  is  the  value  of  .085  of  a  £? 

11.  What  is  the  value  of  .86  of  a  cwt.  ? 

12.  What  is  the  value  of  f  of  .86  cwt.  ? 

13.  What  is  the  value  of  .82  of  a  day? 

14.  What  is  the  value  of  1.089  miles  ? 

15.  What  is  the  value  of  .09375  of  a  pound  Avoirdupois  ? 

16.  What  is  the  value  of  .28493  of  a  year  of  365  days? 
1Y.  What  is  the  value  of  £1.046  ? 

18.  What  is  the  value  of  £1.88  ? 

CASE     V. 

212.  To  reduce  a  compound  number  to  a  common  frac- 
tion of  a  given  denomination. 

1.  Reduce  9  oz.  12  pwt.  to  the  fraction  of  a  pound  Troy. 

OPERATION. 
ozu   pwt  I  Ik. 

ANALYSIS. — In  9  oz.  12  pwt.  there  9     12  10 

are  192  pwt.     In  1  Ib.  there  are  240  20 

pwt.     Therefore,  the  part  of  a  pound         ~~7  1 2 

is  expressed  by  ||| :     Hence,  192  Pwt-  20 

240  pwt. 

Jfjf  -  |  Ib.  Ana. 
Rule. 

Reduce  the  compound  number  to  the  lowest  denomination 
named  in  it,  and  divide  the  result  by  the  number  of  units 
of  that  denomination  which  make  1  of  the  given  denomi- 
nation. 

Examples. 

1.  What  part  of  a  tun  of  wine  is  3  hhd.  31  gal.  2  qt.  ? 

2.  Reduce  3  gal.  2  qt.  to  the  fraction  of  a  hogshead. 

3.  Reduce  2  fur.  36  rd.  2  yd.  to  the  fraction  of  a  mile. 

4.  What  part  of  a  £  is  5s.  TJd.? 

5.  What  part  of  a  pound  Troy  is  10  oz.  13  pwt.  8  gr.  ? 

6.  11  cwt.  0  qr.  12  Ib.  7  oz.  1J-  dr.,  is  what  part  of  a  ton? 


212.   What  is  Case  V.  ?    Give  the  rule. 


REDUCTION.  177 

7.  Reduce  2  R.  32  P.  8  yd.  to  the  fraction  of  an  acre. 

8.  Reduce  12s.  9d.  1J  far.  to  the  fraction  of  a  guinea. 

9.  What  part  of  a  cwt.  is  9  tenths  of  a  pound  ? 

10.  What  part  of  an  Ell  English  is  3  qr.  3  na.  1J  in.  ? 

11.  Reduce  3°  15'  18|"  to  the  fraction  of  a  sign. 

12.  Reduce  3£  inches  to  the  fraction  of  a  hand. 

13.  Reduce  5  yd.  2  ft.  9  inches  to  the  fraction  of  a  mile-. 

CASE    VI. 

213.    To  reduce  a  compound  number  to  a  decimal   of  a 
given  denomination. 

1.  Reduce  ^1  4s.  9|d.  to  the  decimal  of  a  £. 

ANALYSIS. — Reduce  the  fd.  to  a  decimal,  OPERATIC. 

and  annex  the  result  to  the  9d.,  and  we  have  3  i   HC  j 

9.75d.     Dividing  9.7od.  by  12  (since  12  pence  *   '    "  '        ' 

=ls.),  and  annexingtiie  quotient  to  the  4s.,  * 

we  have  4.8125s.    Then  dividing  by  20  (since  12  )  9.75d. 

20s.  =  £1),  and  annexing  the  quotient  to  the  on  N  .  Cio^ 

£1,  we  have  £1.240625:  '  )4.81J58._ 

Ans.     £1  4s.  9  jd.  =  l. 240625  JB. 

Rule. 

I.  If  the  lowest  denomination  contains  a  fraction,  reduce 
it  to  a  decimal,  and  annex  the  integral  part  : 

II.  Then  divide  by  the  scale,  and  annex  the  quotient  as 
a  decimal,  to  the  next  higher  denomination,  and  so  on  until 
the  decimal  is  reduced  to  the  required  denomination. 

Examples. 

1.  Reduce  4  wk.  6  da.  5  hr.  30  m.  45  s.  to  the  decimal  of 
a  week. 

2.  Reduce  2  Ib.  5  oz.  12  pwt.  16  gr.  to  the  decimal  of  a 
pound. 

3.  Reduce  3  feet  9  inches  to  the  decimal  of  yards. 

218.  What  is  Case  VI.?    What  is  the  rule? 
8* 


178  COMPOUND   NUMBERS. 

4.  Reduce  1  Ib.  12  dr.,  avoirdupois,   to   the  decimal  of 
pounds. 

5.  Reduce  5  leagues  2  furlongs  to  the  decimal  of  leagues. 

6.  Reduce  4  bu.  3  pk.  1  pt.  to  the  decimal  of  bushels. 

7.  Reduce  5  oz.  13  pwt.  12  gr.  to  the  decimal  of  a  pound. 

8.  Reduce  15  cwt.  3  qr.  2J  Ib.  to  the  decimal  of  a  ton. 

9.  Reduce  5  A.  3  R.  21  sq.  rd.  to  the  decimal  of  acres. 

10.  Reduce  11  pounds  to  the  decimal  of  a  ton. 

11.  Reduce  3  da.  12f  sec.  to  the  decimal  of  a  week. 

12.  Reduce  14  bu.  3|  qt.  to  the  decimal  of  a  chaldron. 

13.  Reduce  7  m.  7  fur.  1  r.  to  the  decimal  of  miles. 

14.  Reduce  15s.  6d.  3.375  far.  to  the  decimal  of  a  pound. 

15.  Reduce  4°  36'.8125  to  the  decimal  of  a  sign. 


ADDITION. 

214.  ADDITION  OF  COMPOUND  NUMBERS  is  the  operation  of 
finding  a  number  equal  to  two  or  more  given  numbers. 

1.  How  many  pounds,  shillings,  and  pence  are  there  in 
£4  8s.  9d.,  £27  14s.  lid.,  and  £156  17s.  lOd.  ? 

ANALYSIS. — Having  written  the  numbers, 
add  the  column  of  pence;   then  30  pence  OPERATION. 

are  equal  to  2  shillings  and  6  pence :  write  £        8-        d. 

down  the  6,  carrying  the  2  to  the  shillings. 
Find  the  sum  of  the  shillings,  which  is  41 ; 
that  is,  2  pounds  and  1  shilling  over.  Write  156  17  10 

down  Is.;  then,  carrying  the  2  to  the  col-       £189       Is.     6d. 
umn   of  pounds,  we   find  their  sum  to  he 
£189  Is.  6d. 

NOTE. — In  simple  numbers,  the  number  of  units  of  the  scale, 
at  any  place,  is  10.  Hence,  we  carry  1  for  every  10.  In  denom- 
inate numbers,  the  scales  vary.  The  number  of  units,  in  passing 
from  pence  to  shillings,  is  12;  hence,  we  carry  one  for  every  12. 
In  passing  from  shillings  to  pounds,  it  is  20 ;  hence,  we  carry  one 
for  every  20.  In  passing  from  one  denomination  to  another,  we 
divide  the  sum  of  each  column  l>y  the  scale,  and  add  the  quotient 
to  the  next  column;  Hence, 


ADDITION.  179 

Rule. 

I.  Write  the  numbers  to  be  added,  so  that  units  of  the 
same  name  shall  stand  in  the  same  column: 

II.  Beginning  with  the  lowest  denomination,  add  as  in 
simple  numbers  ;  divide  the  sum  of  each  column  by  the  scale, 
and  add  the  quotient  to  the  next  column. 

PROOF. — The  same  as  in  simple  numbers. 

Examples. 

(1.)  (2.) 

£  8.        d.  Ib.  oz.     pwt       gr. 

173  13  5  171  6  13  14} 

87  17  7|  391  11  9  12 

75  18  7£  230  6  6  13 

25  17  8J  94  7  3  18f 

10  10  101  42  10  15  20 

373     18     3 

(3.)  (4.) 

Hi          5       3      3       gr.  T.       cwt     qr.      Ib.         ot 

24       7     2     1     16  15  12  1  10  10J 

17     11     7     2     19  71  8  2  6  0- 

36       6     5     0       7  83  19  3  15  5 

15       9     7     1     13  36  7  0  20  14| 

93419  47  11  2  2  11 


(5.)  *  (6.) 

tun.     pt.    hhd.    gal.  qt  ch.       bu.     pk.    qt     pt 

14  2     1     27  3  27     25     3     7     1 

15  1     2     25  2  59     21     2     6     3 
4     2     1     27  1  21271 
501     62  3  59182 
712     21  2  44       7351 


214.  What  is  Addition  of  Compound  Numbers  ?  How  do  you  set 
down  the  numbers  for  addition  ?  How  do  you  add  ?  What  is  the 
rule  for  addition  ?  How  do  you  prove  addition  ? 


180  COMPOUND   NUMBERS. 

(»•)  (8-) 

yr.  mo.  wk.  da.  hr.  &.  °  '  " 

4  11  3  6  20f  5  17  36  29 
3  10  2  5  21f  7  25  41  21 

5  8  1  4  19T43  8  15  16  09 
J.01  9  3  7  23  3  12  08  10 

55  8  4  6  17  4  17  50  40 


Applications. 

1.  Add  46  Ib.  9  oz.  15  pwt.  16  gr.,  87  Ib.  10  oz.  6  pwt. 
14  gr.,  100  Ib.  10  oz.  10  pwt.  10  gr.,  and  56  Ib.  3  pwt.  6  gr. 
together. 

(2.)  (3.) 

L.      mi.    fur.     rd.      yd.      ft  E.  Fl.     qr.    na. 

16  2  7  39  9  2-J-  126  4  4 

327  1  2  20  7  If  65  3  1 

87  0  1  15  6  1£  72  1  3 

1  1  1  1  2  2|  157  2  3 


(4-)  (5.) 

cu.  yd.     en.  ft.     cu.  in.                                  M.  A.      E.  P.  sq.  yd. 

65       25     1129                       2  60     3  37  25 

37       26       132                       6  375     2  25  21 

50         1     1064                       7  450     1  31  20 

22       19         17  11  30^0  25  19 


6.  What   is   the  weight  of  forty-six  pounds,  eight  ounces, 
thirteen   pennyweights,  fourteen  grains  ;   ninety-seven  pounds, 
three  ounces ;  and  one  hundred  pounds,  five  ounces,  ten  pen- 
nyweights, and  thirteen  grains  ? 

7.  Add  the  following  together:   29  T.  16  cwt.  1  qr.  14  Ib. 
12  oz.  9  dr.,  18  cwt.  3  qr.  1  Ib.,   50  T.  3  qr.  4  oz.,  and  2  T. 
1  qr.  14  dr. 

8.  What  is  the  weight  of  39  T.  10  cwt.  2  qr.  2  Ib.  15  oz 
12  dr.,  17  cwt.  6  Ib.,  12  cwt.  3  qr.,  and  2  qr.  8  Ib.  9  dr.  ? 


APPLICATIONS.  181 

9.  What  is  the  sum  of  the  following:  314  A.  2  R.  39  P. 
200  sq.  ft.  136  sq.  in.,  16  A.  1  R.  20  P.  10  sq.  ft.,  3  R.  36  P., 
and  4  A.  1  R.  16  P.? 

10.  Find  the  contents  of  64  T.  33  ft.  800  in.,  9  T.  1200  in., 
25  ft.  TOO  in.,  and  95  T.  31  ft.  1500  in.  of  round  timber. 

11.  Add  together,  96  bu.  3  pk.  2  qt.  1  pt.,  46  bu.  3  pk. 
1  qt.  1  pt.,  2  pk.  1  qt.  1  pt.,  and  23  bu.  3  pk.  4  qt.  1  pt. 

12.  What   is    the   area   of   the    four   following   pieces   of 
land:   the  first  containing  20  A.  3  R.  15  P.  250  sq.  ft.  116 
sq.  in. ;  the  second,  19  A.  1  R.  39  P. ;  the  third,  2  R.  10  P. 
60  sq.  ft. ;  and  the  fourth,  5  A.  6  P.  50  sq.  in.  ? 

13.  A  farmer  raised  from  one  field,  37  bu.  1  pk.  3  qt.  of 
wheat;  from  a  second,  41  bu.  2  pk.  5  qt.  of  barley;  from  a 
third,  35  bu.  1  pk.  3  qt.  of  rye ;  from  a  fourth,  43  bu.  3  pk. 

I  qt.  of  oats :  how  much  grain  did  he  raise  in  all  ? 

14.  A  grocer  received  an  invoice  of  4  hhd.  of  sugar :  the 
first  weighed  llcwt.  151b. ;  the  second,  12cwt.  3  qr.  151b. ; 
the  third,  9  cwt.  1  qr.  16  Ib. ;  the  fourth,  12  ewt.  1  qr. :  how 
much  did  the  four  weigh  ? 

15.  A  lady  purchased  32  yd.  3  qr.  of  sheeting ;  31  yd.  1  qr. 
of  shirting ;   14  yd.  2  qr.  of  linen ;   and  6  yd.  2  qr.  of  cam- 
bric :  what  was  the  whole  number  of  yards  purchased  ? 

16.  Purchased   a   silver   teapot,  weighing   23  oz.   17  pwt. 

II  gr.  ;  a  sugar-bowl,  weighing  8  oz.  13  pwt.  19  gr. ;  a  cream 
pitcher,  weighing  5  oz.  11  gr. :  what  was  the  weight  of  the 
whole  ? 

17.  A  stage  goes  one  day,  87  m.  6  fur.  24  rd. ;  the  next, 
75  m.   3  fur.  17  rd. ;    the   third,    80  m.  7  fur.  10  rd. ;    the 
fourth,  78  m.  5  fur. :  how  far  does  it  go  in  the  four  days  ? 

18.  Bought  three  pieces  of  land:   the  first  contained  17 
acres  1  R.  35  P.  ;   the   second,   36  acres  2  R.  21  P.  ;   and 
the  third,  46  acres  OR.  37  P.  :   how  much  land  did  I  pur- 
chase ? 


182 


DENOMINATE   FRACTIONS. 


ADDITION     OF     FRACTIONS. 

215.    1.  Add  £f  to  f  s. 

£f  _  2  of  ^o  =  _4_o  shilling. 

Then,     -VL  +  f  =  W  +  M  =  -W-s-  =  ¥-s.  =  14s.  3d. 
Or,  f  s.  may  be  reduced  to  the  fraction  of  a  £:   thus, 

f  s.  =  f  of  &£  =  T!T  of  a  ^  =  2T  of  a  JB. 
Then,  *  +  &  =  «  +  T\  -  f  i  of  a  £, 

which,  being  reduced,  gives  14s.  2d.  .4ns. 

2.    Add  f  of  a  year,  J  of  a  week,  and  %  of  a  day. 

|  of  a  year  =  f  of  *f*  days  =  31  wk.  2  da. 
J  of  a  week  =  £  of  7  days      =      .        2  da.     8  hr. 
•J-  of  a  day       ....      =       ...      3  hr. 

Ans.  31  wk.  4  da.  11  hr. 

Rule. 

Eeduce  the  fractions  to  the  same  unit,  and  then  add  as 
in  simple  fractions. 

Or  :  Eeduce  the  fractions,  separately,  to  integers  of  lower 
denominations,  and  then  add  as  in  denominate  numbers. 

Examples. 

1.  Add  $  of  a  ton  and  T9Q  of  a  cwt. 

2.  Add  f  of  a  yard,  f  of  a  foot,  and  j-  of  a  mile. 

3.  Add  H  miles,  /^  of  a  furlong,  and  30  rods. 

4.  Add  f  cwt.,  -422-  lb.,  13  oz.,  1  cwt.,  and  6  Ib. 

5.  Add  5f  days  and  52rS9  minutes. 

6.  Add  £f,  3.75s.,  and  .975d. 

7.  Add  .2965  T.,  .8725  cwt.,  .63725  qr.,  and  .1625  lb. 

216.  What  is  tfce  rule  for  the  addition  of  denominate  fractions  ? 


SUBTRACTION.  188 

8.  A  tailor  bought  3  pieces  of  cloth,  containing  respect- 
ively, 18f  yards,  21J  Ells  Flemish,  and  16£   Ells   English: 
how  many  yards  in  all  ? 

9.  Mr.  Merchant  bought  of  farmer  Jones,  22-J-  bushels  of 
wheat   at   one   time,   19T55  bushels   at  another,  and   33§   at 
another :  how  much  did  he  buy  in  all  ? 

10.  Mr.  Warren  pursued  a  bear  for  three  successive  days : 
the  first  day  he  traveled  28f  miles ;  the  second,  33-^  miles ; 
the  third,  29^  miles,  when  he  overtook  him :  how  far  had 
he  traveled  ? 

11.  Bought  3  kinds  of  cloth:   the  first  contained  ^  of  3 
of  I  of  -|  yards ;   the   second,  i  of  f  of  5  yards ;   and  the 
third,  ^  of  f  of  £  yards  :  how  much  in  them  all  ? 

12.  Add  11  cwt.,  17  J  lb.,  and  7f  oz. 


SUBTRACTION. 

216.  The  principles  on  which  Subtraction  of  Compound 
Numbers  is  founded,  are  the  same  as  those  that  govern  the 
subtraction  of  simple  numbers. 

1.  What  is  the  difterence  between  £27  16s.  8d.  and  £19 
17s.  9d.? 

ANALYSIS.— We  cannot  take  9d.  from  8d. ;  OPERATION. 

we  therefore  add  to  the  upper  number  as  ^      20    12 

many  units   as   are  contained  in  the   scale,  Ks*       ' 

and  at  the  same  time  add  1,  mentally,  to  the 

next  higher  denomination  of  the  subtrahend.  7      18    11 

We  then  say,  9  from  20,  leaves  11.  Then, 
as  we  cannot  subtract  18  from  16,  we  add  20,  and  say,  18  from 
36,  leaves  18.  Now,  as  we  have  taken  1  pound  =  20  shillings, 
from  the  pounds,  and  added  it  to  the  shillings,  there  are  but  26 
pounds  left.  We  may  then  say,  19  from  26,  leaves  7,  or  20  from 
27,  leaves  7.  The  latter  is  the  easiest  in  practice.  The  first  step 
is  called  ~borrowing ;  the  second,  carrying :  Hence 


184 


COMPOUND   NUMBERS. 


Rule. 

I.  Set  down  the  less  number  under  the  greater,  placing 
units  of  the  same  value  in  the  same  column. 

II.  Begin  with  the  lowest  denomination,  and  subtract  as 
in  simple  numbers,  borrowing  and  carrying  when  necessary, 
according  to  the  scale. 

PROOF. — The  same  as  in  simple  numbers. 

Examples. 


(1.) 

(2.) 

A.       E.       P. 

T.     cwt     qr.      ft. 

From 

18     3     28 

4       12      3      20 

Take 

15     2     30 

2     18     3       1 

Remainder, 

(3.) 

(4.) 

lb.          oz.      pwt      gr. 

lb.        oz.     pwt 

gr. 

From 

273       0       0       0 

18       9     10 

0 

Take 

97     10     18     21 

9     10     15 

20 

Remainder, 

(5.) 

(6.) 

T.     cwt     qr.     lb.       oz. 

cwt     qr.      lb.       oz. 

dr. 

From 

7     14     1     3       6 

14     2     12     10 

8 

Take       . 

2       6     3     4     11 

6     3     16     15 

3 

Remainder, 

a) 

(8.) 

tun. 

hhd.    gal.      qt     pt                yr. 

wk.      da.     hr.      min 

sec. 

From     151 

3     50     3     2             95 

25     4     20     45 

50 

Take       27 

2     54     3     2             80 

30     6     23     46 

56 

Rem., 

(9.) 

(10.) 

mi.     fur. 

rd.      yd.    ft. 

A.        E.       P.     sq.  yd 

.  sq.  ft 

From     84     7 

13     2     1           From 

145     3     35     19 

H 

Take      69     7 

25     4     2|          Take 

98     3     39     25 

85 

216.  What  is  the  rule  for  Subtraction  of  Compound  Numbers  ? 


SUBTRACTION. 


185 


11.) 
£         B.         d. 

From       25     17     4£ 
Take        19     19     9i 


(18.) 

From       11     24     19 


Take 


9     29     59     59J 


13.  From  38  mo.  2  wk.  3  da.  7  hr.  10  m.,   take  10  mo. 
3  wk.  2  da.  10  hr.  50  m. 

14.  From  176  yr.  8  mo.  4  wk.  4  da.,  take  91  yr.  9  mo. 
2  wk.  6  da. 

15.  From  ^3,  take  3s. 

16.  From  2  lb.,  take  20  grr  Troy. 

17.  From  8  ft,  take  1»  15  23  23. 

18.  From  9  T.,  take  1  T.  1  cwt.  2  qr.  20  lb.  15  oz.  14  dr. 

19.  From  3  miles,  take  3  fur.  19  rd. 

20.  I  purchased  167  lb.  8  oz.  16  pwt.  10  gr.  of  silver,  and 
sold  98  lb.  10  oz.  12  pwt.  19  gr.:  how  much  had  I  left  ? 

21.  I  bought  19  T.  11  cwt.  2  qr.  2  lb.  12  oz.  12  dr.  of  old 
iron,  and  sold  17  T.  13  cwt.  2  qr.  19  lb.  14  oz.  10  dr.:  what 
had  I  left  ? 

22.  I  purchased    101  ft   11 1  73   23  19  gr.  of  medicine, 
and    sold    17  ft    23    33    13    5  gr. :     how    much    remained 
unsold  ? 

23.  From  46  yd.  1  qr.  3  na.,  take  42  yd.  3  qr.  1  na.  2  in. 

24.  Bought  7  cords  of  wood,  and  2  cords  78  feet  having 
been  stolen,  how  much  remains  ? 


TIME     BETWEEN     DATES. 
217.    To  find  the  time  between  any  two  dates. 

1.  What  time  elapsed  between  July  5th,  1848,  and  August 
8th,  1850? 

NOTE. — In  the  first  date,  the  number  of  the 
year  is  1848 ;  the  number  of  the  month,  V, 
and  the  number  of  the  day,  5.  In  the  second 
date,  the  number  of  the  year  is  1850,  the 
number  of  the  month,  8,  and  the  number  of 
the  day,  8.  Hence,  to  find  the  time  between 
two  dates: 


OPERATION, 
yr.        mo.    da 

1850     8     8 

1848     7     5 

213 


186  COMPOUND    NUMBERS. 

Rule, 

Write  the  numbers  of  the  earlier  date  under  those  of  the 
later,  and  subtract  according  to  the  preceding  Hide. 

NOTES.  —  1.  In  finding  the  difference  between  dates,  as  in  cas*-- 
ing  interest,  tbe  month  is  regarded  as  the  twelfth  part  of  a  year, 
and  as  containing  30  days. 

2.   The  civil  day  begins  and  ends  at  12  o'clock  at  night. 

2.  What  is  the  difference  of  time  between  March  2d,  1847, 
and  July  4th,  1856  ? 

3.  What  is  the  difference  of  tune  between  April  28th,  1834, 
and  February  3d,  1856  ? 

4.  What  time  elapsed  between  November  29th,  1836,  and 
January  2d,  1854  ? 

5.  What  time  elapsed  between  November  8th,  at  11  o'clock, 
A.  M.,   1847,  and  December  16th,  at  4  o'clock,  p.  M.,  1850? 


ANALYSIS.—  The   hours   are   numbered 
from  12  at  night,  when  the  civil  day  be- 

mi  7  /»  ,1  ,T 

gins.     The  numbers  of  the  years,  months,         1Q,*     ,,        Q     ,., 

•%  it  -i  -tOTci.Ll.OjLJ. 

days,  and  hours,  are  used. 

3185 

6.  What  time  elapsed  between   October  9th,  at  11  p.  M., 
1840,  and  February  6th,  at  9  P.  M.,  1853  ? 

7.  Mr.  Johnson   was    born    September    6th,    1771,    at    9 
o'clock,  A.  M.,  and  his  first  child  November  5th,  1801,  at  9 
o'clock,  p.  M.  :  what  was  the  difference  of  their  ages  ? 

8.  The   revolution   commenced   April    19th,    1775,    and   a 
general  peace  took  place  January  20,  1783  :   how  long  did 
the  war  continue  ? 

9.  America   was    discovered    by    Columbus,    October    11, 
1492  :  what  was  the  length  of  time  to  July  25,  1862  ? 

217.  Give  the  rule  for  finding  the  difference  between  two  dates. 
How  is  the  month  reckoned?  At  what  time  does  a  civil  day 
begin  ? 


DENOMINATE    FRACTIONS.  187 

SUBTRACTION     OF    FRACTIONS. 

1.  From  ^  of  a  £,  take  J  of  a  shilling. 

J  of  a  shilling  =  J  of  ^  of  a  ^  =  ^  of  a  £. 
Then,  J  jg-  ^dB  =  f  g  -  ^  =  f §  of  a  £  =  9s.  8d. 

2.  From  If  Ib.  Troy  weight,  take  J-  oz. 

lb.     oz.       pwt       gr. 

If  lb.  =  J  lb.  of  Jr2-  oz.  =    1     9 
J  oz.  =  $  of  -2r°-  of  Y-  gr.  =  80  gr.  =    0    "0       3       8 

-4ns.    1     8     16     16. 
Hence,  the  following 

I  Rule. 

Reduce  the  fractions  to  the  same  unit,  and  then  subtract 
as  in  simple  fractions, 

Or :  Eeduce  the  fractions,  separately,  to  integers  of  lower 
denominations,  and  then  subtract  as  in  compound  numbers. 

Examples. 

1.  From  f  oz.,    take   J  pwt. 

2.  From  £1$,    take   f   of  a   shilling. 

3.  From  1}  oz.,    take   f-  pwt. 

4.  From  i   of  a  day,   take  J  of  a  second. 

5.  From  J   of  a  rod,   take   ^   of  an   inch. 

6.  From  T4?   of  a   hogshead,   take   f   of  a  quart. 

7.  From  f  oz.,   take  J  pwt. 

8.  From  4f  cwt.,   take   4T%  lb. 

9.  From  8f  cwt.,   take   4T9o  lb. 

10.  From   3i  lb.  Troy  weight,   take   %  oz. 

11.  From   l-i-  rods,    take   f   of  an  inch. 

12.  From   -ff  fo,    take    T7g  5. 

13.  From  .69875  of  a  tun,  take  .386125  hhd. 

14.  From  2.9675  wk.,  subtract  5.96974  days. 

15.  From   f  cwt.   of  sugar,  there   was   taken   .2125  qr. : 
what  was  the  remainder  worth,  at  6  cents  a  pound  ? 


188  COMPOUND   NUMBERS. 


MULTIPLICATION. 

218.  MULTIPLICATION  OF  COMPOUND  NUMBERS  is  the  opera- 
tion of  taking  a  compound  number  as  many  times  as  there 
are  units  in  the  multiplier, 

A  tailor  has  5  pieces  of  cloth,  each  containing  6  yd.  2  qr 
3  na.  :  how  many  yards  are  there  in  all  ? 

ANALYSIS. — In  all  the  pieces  there  are  5  times        OPERATION. 
as  much  as  there  is  in  1  piece.     If,  in  1  piece,      yd.      qr.      na. 
each  denomination  be  taken  5  times,  the  result       623 

will  be  5  times  as  great  as  the  multiplicand. 5 

Taking  each  denomination  5  times,  we  have  30     30     JQ     ^5 
yd.  10  qr.  15  na.  33        1        3 

But,  instead  of  writing  the  separate  products, 
we  begin  with  the  lowest  denomination  and  say,  5  times  3  na. 
are  15  na. ;  divide  by  4,  the  units  of  the  scale,  write  down  the 
remainder,  3  na.,  and  reserve  the  quotient.  3  qr.,  for  the  next 
product.  Then  say,  5  times  2  qr.  are  10  qr.,  to  which  add  the 
3  qr.,  making  13  qr.  Then  divide  by  4,  write  down  the  remain- 
der, 1,  and  reserve  the  quotient,  3,  for  the  next  product.  Then 
say,  5  times  6  are  30,  and  3  to  carry,  are  33  yards:  Hence, 

Rule. 

I.  Write  down  the  denominate  number,  and  set  the  mul 
tiplier  under  the  lowest  denomination: 

II.  Multiply  as  in  simple  numbers,  and  in  passing  from 
one  denomination   to    another,   divide   by  the  units  of  the 
scale,  set  down  the  remainder,  and  carry  the  quotient  to  the 
next  product. 

PROOF. — The  same  as  in  simple  numbers. 

Examples. 
(I-)  (2-) 

£         s.  d.     far.  T.    cwt.     qr.        Ib.      oz. 

17     15       9     3  10     0       2     12 

6  7 


106     14     10     2  3     10     0     19 


MULTIPLICATION.  189 

(3.)  (4.) 

m.    fur.     rd.     yd.     ft  s.         °  '          " 

9     3     20     3     2  9       9       27     35 

6  3 


(5.)  (6.) 

yr.    mo.     da.       hr.  T.      cwt    qr.       Ib.  oz.      dr. 

6     5     15     18  6     12     3     20  12  '  9 
5                                                    8 


7.   How  much  sugar  in  12  barrels,  each  containing  3  cwt. 
3  qr.  2  Ib. 

OPERATION. 

T.      cwt       qr.        Ib. 

ANALYSIS. — The    multiplicand    is    3    cwt.  332 

3  qr.  2  Ib. ;    and    the  multiplier  12,   a   com-  3 

posite  number :  we  therefore  multiply  by  3  ~ 
and  4,  in  succession. 


2       5       0     24 

8.  A  farmer  has  11  bags  of  corn,  each  containing  2  bu. 
1  pk.  3  qt.  :    how  much  corn  in  all  the  bags  ? 

9.  In  7  loads  of  wood,  each  containing  1  cord  and  2  cord 
feet,  how  many  cords  ? 

10.  A  bond  was  given  21st  of  May,  1825,  and  was  taken 
up  the  12th  of  March,  1831  :    what  will  be  the  product,  if 
the  time  which  elapsed  from  the  date  of  the  bond  till  the 
time  it  was  taken  up  be  multiplied  by  3  ? 

11.  What  is   the  weight  of  1  dozen  silver  spoons,   each 
weighing  3  oz.  6  pwt.  ? 

12.  What  is  the  weight  of  7  tierces  of  rice,  each  weighing 
5  cwt.  2  qr.  161b.? 

13.  Bought  4  packages  of  medicine,  each  containing  3  Ib. 
4  §  63  1  3  16  gr. :    what  is  the  weight  of  all  ? 

14.  How  far  will  a  man  travel  in  5  days,  at  the  rate  of 
24  mi.  4  fur.  4|  rd.,  per  day? 

218.   What  is  Multiplication  of  Compound  Numbers?     What  is 
the  rule? 


190 


COMPOUND    NUMBERS. 


15.  Hew  much  land   is  there  in  9  fields,  each   field  con- 
taining 12  A.  1  R.  25  P.? 

16.  How  many  yards  in  9  pieces,  each  29  yd.  2  qr.  3  na.  ? 

17.  If  a  vessel  sails  5  L.  2  mi.  6  fur.  36  rd.  in  one  day, 
how  far  will  it  sail  in  8  days  ? 

18.  How  much  water  will  be  contained  in  96  hogsheads, 
each  containing  62  gal.  1  qt.  1  pt.  1  gi.  ? 

19.  If  one  spoon  weighs  3  oz.  5  pwt.  15  gr.,  what  is  the 
weight  of  120  spoons  ? 

20.  If  a  man  travels  24  mi.  7  fur.  4  rd.  in  one  day,  how  far 
will  he  go  in  one  month  of  30  days? 

21.  If  the  earth  revolves  0°  15'  of  space   per  minute  of 
time,  how  far  does  it  revolve  per  hour? 

22.  Bought  90  hhd.  of  sugar,  each  weighing  12  cwt.  2  qr. 
11  Ib. :   what  was  the  weight  of  the  whole  ? 

23.  What  is  the  cost  of  18  sheep,  at  5s.  9|d.  apiece  ? 

24.  How   much   molasses   is   contained   in   25  hhd.,    eacb 
hogshead  having  61  gal.  1  qt.  1  pt.  ? 

25.  How  many  yards  of  cloth  in   36   pieces,   each  piec1 
containing  25  yd.  3  qr.  ? 


DIVISION. 


219.  DIVISION  OP  COMPOUND  NUMBERS  is  the  operation  of 
finding  how  many  times  one  number  contains  another,  T/hen 
one  or  both  are  compound. 

1.    Divide  ^£25  15s.  4d.  by  8. 

ANALYSIS. — We  first  say,  8  into  25, 

3  times  and  £1  or  20s.  ovev.     Then,  OPERATION". 
after  adding  the  15s.,  we  say,  8  into  35,            8  )  £25     15s.     4d. 

4  times  and  3s.  over.     Then,  reducing  -,„ "        77 
the  3s.  to  pence,  and  adding  in  the  4d.,  ^ 
we  say,  8  into  40,  5  times. 


DIVISION.  191 

2.  Divide  36  bu.  3  pk.  7  qt.  OPERATION. 

by  7.  7  )  36  bu.  3  pk.  7  qt.  (  5  bu. 

o- 

ANALYSIS. — In  this   example,  _ 

we  find  that  7  is  contained  in  1 

36  bushels,  5  times  and  1  bushel  4 

over.     Reducing  this  to  pecks,  ^  ~    ,     .        , 

and    adding    3    pecks,    gives    7  ',  V  1 1 

pecks,  which  contains  7,  1  time  _ 

and  no  remainder.    Multiplying  0 
0  by  8  quarts,  and  adding,  gives 

7  quarts  to  be   divided  by  7 :  7  VI  (  1  qt 
Hence,  when  the  divisor  is  an 
abstract  number,  we  have  the 

following  Ann.  5  bu.  1  pk.  1  qt. 
Rule. 

1.  Begin  with  the  highest  denomination   and  divide  as 
in  simple  numbers: 

II.  Reduce  the  remainder,  if  any,  to  the  next  lower  de- 
nomination  and  add  in  the  units  of  that  denomination,  for 
a  new  dividend: 

III.  Proceed  in  the  same  manner,  through  all  the  denom- 
inations. 

PROOF. — By  multiplication,  as  in  simple  numbers. 

NOTES. — 1.  If  the  divisor  is  a  composite  number,  we  may  divide 
by  the  factors  in  succession,  as  in  simple  numbers. 

2.  Each  quotient  figure  has  the  same  unit  as  the  dividend  from 
which  it  was  derived. 

3.  If  the  divisor  is  a  denominate  number,  reduce  it  and  the  divi- 
dend to  the  same  unit,  and  then  divide  as  in  simple  numbers. 

Examples. 

(1.)  (20 

T.      cwt     qr.       Ib.  A.         E.       P. 

7)1     19     2     12  9)113     3     25 

Quotient,  5     2     16 

(3.)  (4.) 

L.     mi.    fur.    rd.  bu.     pk.     qt 

8  )*T     *     T     8  9  )25     3    4 

Quotient, 


192 


COMPOUND   NUMBERS. 


5.  Divide  17  cwt.  0  qr.  2,lb.  6  oz.  by  7. 

6.  Divide  49  yd.  3  qr.  3  na.  by  9. 

7.  If  a  man,  lifting  8  times  as  much  as  a  boy,  can  raise 
201  Ib.  12  oz.,  how  much  can  the  boy  lift  ? 

8.  If  a  vessel  sails  25°  42'  40"  in  10  days,  how  far  will 
she  sail  in  one  day? 

9.  Divide  9  hhd.  28  gal.  2  qt.  by  12. 

10.  What  is   the   quotient  of  65  bu.  1  pk.  3  qt.  divided 
by  12? 

11.  In  4  equal  packages  of  medicine,  there  are  13  RJ  7  % 
23  13  4  gr. :  how  much  is  there  in  each  package  ? 

12.  In  9  fields  there  are  113  A.  3  R.  25  P.  of  land:   if 
the   fields  contain  an  equal  amount,   how  much  is   there  in 
each  field? 

13.  If  15  loads  of  hay  contain  35  T.  5  cwt.,  what  is  the 
weight  of  each  load  ? 

14.  In  25  hhd.  of  molasses,  the  leakage  has  reduced  the 
whole  amount  to  1534  gal.  1  qt.  1  pt. :  if  the  same  quantity 
has  leaked  out  of  each  hogshead,  how  much  will  each  hogs- 
head still  contain  ? 

15.  Bought  65  yards  of  cloth,  for  which  I  paid  £72  14s. 
4|d. :  what  did  it  cost  per  yard  ? 


16. 
17. 


£1138  12s.  4d.^53. 


18.  70  T.  17  cwt.  71b.~  79. 


27  bu.  Tqt.-f-84.  19.  1 14  hhd.  5 6  gal.  1  qt.-^  40. 


20.  If,  in  30  days,  a  man  travels  746  mi.  5  fur.,  traveling 
the  same  distance  each  day,  what  is  the  length  of  each  day's 
Journey ? 

.  21.  Suppose  a  man  had  98  Ib.  2  oz.  19  pwt.  5  gr.  of  sil- 
ver :  how  much  must  he  give  to  each  of  7  men,  if  he  divides 
it  equally  among  them  ? 

22.  When  175  gal.  2  qt.  of  beer  are  drank  hi  52  weeks, 
how  much  is  consumed  in  one  week  ? 


219.  What  is  Division  of  Compound  Numbers  ?  Give  the  rule 
for  division.  How  do  you  prove  division  ?  How  do  you  divide, 
when  the  divisor  is  a  composite  number?  What  will  be  the  unit 
of  each  quotient  figure? 


APPLICATIONS.  193 


Applications  in  the  foregoing  Rules. 

1.  A   farmer   has    18   lots,  each  of  which  contains  41  A. 

2  R.  11  P.;   these   are   divided  among  his  7  children:   how- 
much  does  each  child  have  ? 

2.  A  rich  man  divided  168  bu.  1  pk.  6  qt.  of  corn,  among 
35  poor  men :  how  much  did  each  receive  ? 

3.  There  are  three  men,  the  sum  of  whose  ages  is  14  times 
20  yr.  5  mo.  3  wk.  6  da. :  if  the  ages  are  equal,  what  is  the 
age  of  each  ? 

4.  In  sixty-three  barrels  of  sugar,  there  are  7  T.  16  cwt. 

3  qr.  12  Ib. :  how  much  is  there  in  each  barrel  ? 

5.  One  hundred  and  seventy-six  men  consumed,  in  a  week, 

13  cwt.  2  qr.  15  Ib.  6  oz.  of  bread :  how  much  did  each  man 
consume  in  1  day  ? 

6.  If  the  earth  revolves  on  its  axis  15°  in  1  hour,  how 
far  does  it  revolve  in  1  minute  ? 

7.  A  farmer  has  a  granary  containing  232  bushels  3  pecks 

7  quarts  of  wheat,  and  he  wishes  to  put  it  in  105  bags :  how 
much  must  each  bag  contain? 

8.  If  59  casks  contain  44  hhd.  53  gal.  2  qt.  1  pt.  of  wine, 
what  are  the  contents  of  J  of  a  cask  ? 

9.  Bought  90  hhd.  of  sugar,  each  weighing   12  cwt.  2  qr. 

14  Ib. :  what  is  the  value  of  the  sugar,  at  6  J  cents  per  Ib.  ? 

10.  If  90  hogsheads  of  sugar  weigh  56  T.  14  cwt.  3  qr. 

15  Ib.,  what  is  the  weight  of  1  hogshead  ? 

11.  If  a  vessel  sails  30  days,  at  the  rate  of  49  mi.  6  fur. 

8  rd.  per  day,  at  what  rate  per  day  does  another  ship  sail, 
that  performs  the  same  voyage  in  12  days  ? 

12.  Divide  £18  6s.  9d.  by  £4  9s.  3d. 

NOTE. — Reduce  both  quantities  to  pence,  and  then  divide. 

13.  A  steamship,  in  crossing  the  Atlantic,  has  a  distance 
of  3500  miles  to  go :  if  she  sails  211  mi.  4  fur.  32  rd.  a  day, 
what  distance,  after  15  days,  has  she  still  to  sail? 

14.  A  printer  uses  one  sheet  of  paper  for  every  16  pages 
of  an   octavo   book :    how  much  paper  will  be  necessary  to 

9 


19±  COMPOUND    NUMBERS. 

print  500  copies  of  a  book  containing  336  pages,  allowing  2 
quires  of  waste  paper  in  each  ream  ? 

15.  How  many  barrels   of  sugar,   containing  2  cwt.  1  qr. 
15  lb.,  can  be  filled  from  a  hogshead  of  1  T.  5  cwt.  2  qr. 
20  lb.  ? 

16.  A  man  lends  his  neighbor  .£135  6s.  Sd.,  and  takes  in 
part  payment,  4  cows,  at  .£5  8s.  apiece,  also  a  horse,  worth 
<£50  :   how  much  remained  due  ? 

17.  If  a  man  travels   24  mi.  1  fur.  30  rd.  in  a  day,  how 
long  will  it  take  him  to  travel  200  mi.  6  fur.  18  rd.  ? 

18.  Out  of  a  pipe  of  wine,  a  merchant  draws  12  bottles, 
each   containing   1  pint   3  gills  ;    he   then   fills   six   5-gallon 
demijohns  ;  then  he  draws  off  3  dozen  bottles,  each  contain- 
ing 1  quart  2  gills :  how  much  remained  in  the  cask  ? 

19.  If  a  barrel  of  flour  costs  .£1  4s.  9d.,  how  many  barrels 
can  be  bought  for  JE2T5  10s.  6d.  ? 

20.  Suppose  a  man  has  246  mi.  6  fur.  36  rd.  to  travel  in 
12  days :  after  traveling  9  days,  how  far  has  he  yet  to  travel? 

21.  A  vessel  arrives  in  port,  with  a  cargo  of  50  hogsheads 
of  sugar,   each  containing   1  T.  5  cwt.  3  qr. ;    40  hogsheads, 
each  containing  18  cwt.  2  qr.  12  lb.,  and  15  hhd.,  each  con- 
taining   15  cwt.  3  qr.  18  lb. :    8   merchants   buy  the   entire 
cargo  :   what  amount  of  sugar  belongs  to  each  ? 

22.  A  ship,  with  a  cargo   of  250   bales  of  cotton,  each 
weighing  12  cwt.  2  qr.  15  lb.,  was  overtaken  by  a  storm,  and 
obliged  to  throw  overboard  60  bales;  the  cotton  was  owned 
in  equal  shares,  by  5  merchants :  what  was  the  loss  to  each, 
at  25  cents  per  lb,  and  what  quantity  did  each  receive  ? 

23.  How  many  pieces  of  cloth,  each  containing  35  yards, 
will  clothe  a  company  of  48  men,  if  it  takes  5  yd.  3  qr.  2  na. 
for  each  man  ? 

24.  A  merchant  bought   15  pieces   of  cloth,   3  of  which 
contained  each  34  yd.  3  qr. ;  6  contained  each  31  yd.  1  qr.  3  na., 
and  the  remainder,  40  yd.  2|  qr.  each  :   allowing  2  yd.  3  qr. 
for  waste,  how  many  suits,  each  requiring  6  yd.  1  qr.  3  na., 
can  be  made  from  the  cloth  ? 


LONGITUDE   AND   TIME.  195 


LONGITUDE    AND    TIME. 

220.  The  Equatorial  Circumference  of  the  Earth  is  divided 
into  360°,  which  are  called  degrees  of  Longitude. 

221.  The  Sun   apparently  goes   round  the   earth  once   in 
24  hours.     This  time  is  called  a  day.     Hence,  in  24  hours, 
the  sun  apparently  passes  over  360°  of  longitude  ;   and  in 
1  hour,  over  ^  of  360°  =  15°. 

222.  Since  the  sun,  in  passing  over  15°  of  longitude,  re- 
quires 1  hour,  or  60   minutes   of  time,  in  1  minute  of  time 
he  will  pass  over  JL.  Of  15°  =  £$°  =  J°  =  15'  of  longitude  ; 
and  in  1  second  of  time,  over  ^ff  of  15'  =  JJ'  =  y  =  15"  of 
longitude  :    Hence,  / 

15°  of  longitude  require    .       .      1  hour  of  time ; 
15'         "  ^      "         "        .       .      1  minute  of  time ; 
15"        "        "      '    "        .       .      1  second  of  time. 

Hence  we  see,  that, 

1.  If  the  longitude,  expressed  in  degrees,  minutes,  and 
seconds,   be  divided  by   15  =  3  x  5^  the  'quotients   will  be 
hours,  minutes,  and  seconds,  of  time. 

2.  If  time,  expressed  in  hours,  minutes,  and  seconds,  be 
multiplied  by  15  =  3  x  5,  the  product  will  be  degrees,  min- 
utes, and  seconds  of  longitude. 

223.  When  the  sun  is  on  the  meridian  of  any  place,  it  is 
12  o'clock,  or  noon,  at  that  place. 

Now,  as  the  sun  apparently  goes  from  east  to  west,  at  the 
instant  of  noon,  at  one  place,  it  will  be  past  noon  for  all 

220.  How  is  the  circumference  of  the  earth  supposed  to  be  di 
vided  ? 

221.  How  does  the  sun  appear  to  move?    What  is  a  day?    How 
far  does  the  sun  appear  to  move  in  1  hour? 

222.  How  do  you  reduce  degrees  of  longitude  to  time  ?    How  do 
you  reduce  minutes   of  longitude  to  time?    How  do  you  reduce 
seconds  to  time  ?    How  do  you  reduce  time  to  longitude  ? 


196  COMPOUND   NUMBERS. 

places  at  the  east  of  it,  and  before  noon  for  all  places  at 
the  west.  Hence,  if  we  find  the  difference  of  time  between 
two  places,  and  know  the  exact  time  at  one  of  them,  the 
corresponding  time  at  the  other  will  be  found  by  adding  this 
difference  to  the  given  time,  if  the  place  be  East,  or  by  sub- 
tracting it,  if  West. 

224.  The  meridian  of  the  Observatory  of  Greenwich,  Lon- 
don, is  the  one  from  which  longitude  is  reckoned  ;  hence,  the 
longitude  of  Greenwich  is  0. 

Longitude  is  estimated  :  West,  180°  ;   and  East,  180°. 

1.  Baltimore  is  in  longitude  76°  37'  west,  and  New  York 
in  longitude  74°  01'  west.  When  it  is  12  IT.  at  Baltimore, 
what  is  the  time  at  New  York  ? 


ANALYSIS.  —  The  difference  of  ^       76°  37' 

longitude  is  2°  36',  and,  changed  15  =  3  X  5       ^     Q1 
to  time  by  dividing  by  15  =3x5, 

gives  10m.  24  sec.  for  the  differ-  •  6)_Z_6b_ 

ence  of  time  ;  and  as  New  York  5  )  52  m. 

is  east  of  Baltimore,  the  time  is 


later,  and  we  add  :'  10  m-  24  sec- 

12  +  10'  +  24"  =  12  hr.  10  m.  24  sec. 

2.  The  longitude  of  New  York  is  74°  1'  west,  and  that 
of  Philadelphia  75°  10'  west:   what  is  the  time  at  Philadel- 
phia when  it  is  12  M.  at  New  York  ? 

3.  The  longitude   of  Cincinnati,    Ohio,    is    84°    24'  west  : 
what  is  the   time   at  Cincinnati,  when  it   is   12  M.   at  Ne\v 
York  ? 

4.  The  longitude   of  New  Orleans  is  89°  2'  west  :   what 
time  is  it  at  New  Orleans,  when  it  is  12  M.  at  New  York  ? 

5.  The  longitude  of  St.  Louis  is  90°  15'  10"  west :  what  is 

223.  What  is  the  hour  when  the  sun  is  on  the  meridian  ?  When 
the  sun  is  on  the  meridian  of  any  place,  how  will  the  time  be  for 
all  places  East  ?  How  for  all  places  West  ?  If  you  have  the  differ- 
ence of  time,  how  do  you  find  the  time  at  either  place? 
~224.  From  what  meridian  is  longitude  reckoned  ?  What  is  the 
longitude  of  this  meridian  ?  How  is  longitude  reckoned  from  it  ? 


LONGITUDE   AND   TIME.  197 

the  time  at  St.  Louis,  when  it  is  3  h.  25  m.,  p.  M.,  at  New 
York  ? 

6.  The  longitude  of  Boston  is  71°  4'  west,  and  that  of 
New  Orleans  89°  2'  west  :  what  is  the  time  at  New  Orleans, 
when  it  is  7  o'clock  12  m.,  A.M.,  at  Boston? 

7.  The   longitude   of  Chicago,    Illinois,    is  87°  30'  west: 
what  is  the  time  at  New  York,  when  it  is  12  M.  at  Chicago  ? 

225.  Knowing  the  difference  of  time  of  two  places,  to 
find  their  difference  'of  longitude. 

1.  Louisville,  in  Kentucky,  is  in  longitude  85°  30'  west, 
and  it  is  9  o'clock,  A.  M.,  at  the  City  of  Mexico,  when  it  is 
9  hr.  54  min.  20  sec.,  A.  M.,  at  Louisville  :  what  is  the  longi- 
tude of  the  City  of  Mexico  ? 

OPERATION. 
br.    min.     see. 

ANALYSIS.—  The  difference  of  time  is  9     54  20 

first  obtained,  which  is  54  min.  20  sec.  9     00  00 

Changing  this  into  longitude  by  multi-  r  *  on 
plying  by  15,  we  have  13°  35'  for  the 
difference    of    longitude.     The    earlier 

time   being  at  the   City  of  Mexico,  it  2     43  00 

must  lie  to  the  west:  hence,  its  longi-  __  5 

tude  is  found  by  adding  the  difference  ^3°  35'  QQ" 

to  the  lonitude  of  Louisville.  O 


99°  05'  Ans. 

2.  Cincinnati  is  in  longitude   84°  21'  west,   and   it  is   10 
o'clock,  A.  M.,  at  Cincinnati,  when  it  is  21  min.  56  sec.  past 
10  at  Buffalo  :   what  is  the  longitude  of  Buffalo  ? 

3.  By  the  chronometer,  it  is  5  hr.  6  min.""  28  sec.,  p.  M.,  at 
Greenwich,  London,  when  it  is  12  M.  at  Baltimore  ;    Green- 
wich is  in  0°  longitude  :  what  is  that  of  Baltimore  ? 

4.  By  the  chronometer,  it  is  4  hr.  56  min.  4^  sec.,  p.  M., 
at  Greenwich,  when  it  is  12  jr.  at  New  York  :   what  is  the 
longitude  of  New  York  ? 

5.  A  captain,  at  sea,  finds  by  his  chronometer,  that  it  ib 
2  hr.  15  min.  30  sec.,  p.  M.,  at  Greenwich,  when  it  is  12  M. 
on  board  his  vessel  :  in  what  longitude  is  the  vessel  ? 


198 


DUODECIMALS. 


DUODECIMALS. 

226.  If  the  unit,  1  foot,  be  divided  into  12  equal  parts, 
each  part  is  called  an  inch,  or  prime,  and  marked,  '.  If  an 
inch  be  divided  into  12  equal  parts,  each  part  is  called  a 
second,  and  marked,  ".  If  a  second  be  divided,  in  like  man 
ner,  into  12  equal  parts,  each  part  is  v  called  a  third,  and 
marked,  '";  and  so  on,  for  divisions  still  smaller. 

The  divisions  of  the  foot,  give 

1'     inch,  or  prime,       .       .       .     =    J_    of  a  foot. 
1"    second  is  T^  of  T^ 
I'"  third  is  TL  of  Ti,  of  A 


—  -j-J—    of  a  foot. 


.     =  -1^  of  a  foot. 

Hence  :  DUODECIMALS  are  denominate  fractions,  in  which 
the  primary  unit  is  1  foot,  and  12  the  scale  of  division. 

NOTES. — 1.  Duodecimals  are  chiefly  used  in  measuring  surfaces 
and  solids. 

2.-  The  marks,  ',  ",  '",  &c.,  which  denote  the  fractional  units, 
are  called,  indices. 

Table. 

12'" make    1"  second. 

12" 1'   inch,  or  prime. 

12' - 1    foot. 

Table  Reversed. 


I       

=     12     = 


1      

12     = 
144     = 


SCALE. — Uniform,  and  equal  to  12. 


12. 
144. 

1728. 


226.  If  1  foot  be  divided  into  twelve  equal  parts,  what  is  each 
part  called  ?  If  the  inch  be  so  divided,  what  is  each  part  called  1 
What  are  Duodecimals  ?  For  what  are  Duodecimals  chiefly  used  1 
What  is  "the  scale? 


MULTIPLICATION.  199 


ADDITION  AND  SUBTRACTION. 

227.   The  units  of  Duodecimals   are  reduced,  added,  and 
subtracted  like  those  of  other  denominate  numbers. 

Examples. 

1.  In  185',  how  many  feet? 

2.  In  250",  how  many  feet  and  inches  ? 

3.  In  4367"',  how  many  feet  ? 

4.  What  is  the  sum  of  3  ft.  6'  3"  2'",  and  2  ft.  1'  10" 
11'"? 

5.  What  is  the  sum  of  8  ft.  9'  7",  and  6  ft.  7'  3"  4'"? 

6.  What  is   the   difference  between  9  ft.  3'  5"  6'",   and 
7ft.  3'  6"  7'"? 

7.  What  is  the  difference  between  40  ft.  6'  6",  and  29  ft. 
7"'? 

8.  What  is  the  difference  between  12  ft.  V  9"  6"',  and 
4ft.  9'  7"  9'"? 

9.  Reduce  6'  8"  to  the  fraction  of  a  foot. 

10.  Reduce  9'  10"  8'"  to  the  fraction  of  a  foot. 

11.  Reduce  4'  5"  3'"  to  the  decimal  of  a  foot. 

12.  Reduce  7"  6'"  to  the  decimal  of  a  foot. 


MULTIPLICATION   OF   DUODECIMALS. 

228.  MULTIPLICATION  OF  DUODECIMALS  is  an  abbreviated 
method  of  finding  the  measure  of  surfaces  or  solids. 

Two  dimensions,  multiplied  together,  produce  square  meas- 
ure ;  and  three  dimensions,  multiplied  together,  produce  cubic 
or  solid  measure. 

227.  How  do  you  add  and  subtract  Duodecimals  ? 

228.  What  is  Multiplication  of  Duodecimals  ?    What  do  two  di 
mensions,  multiplied  together,  produce  ?    Three  dimensions  ?    Feet 
multiplied  by  feet,   give  what  ?     Primes  by  primes  ?     Primes  by 
seconds  ?    Seconds  by  seconds  ?    Primes  by  feet  ?    What  index  must 
a  product  have  ?    Give  the  rule. 


200 


DUODECIMALS. 


The  multiplication  of  duodecimals  is  governed  by  the  fol- 
lowing principles  : 

1.  Feet  multiplied  by  feet,  give  square  feet. 

2.  Primes  x  Primes  =  y1^  ft.  x  T^  ft.  =  TJ-T  ft,,  or  seconds. 

3.  Primes  x  Feet  =  ft  ft.  x*  1  ft.  =  T^  ft.,  or  primes. 

4.  Primes  x  Seconds  =  ft  ft.  x  ^  —  T  ft  ?  ft.,  or  thirds. 


5.   Seconds  x  Seconds  =        ft.  x 


ft.  —  20*a6,  or  fourths. 


OPERATION. 
6  ft.  V  8" 
2  ft.  9' 

4  ft.  11'  9"  0'" 
13          3;  4" 

18  ft.     3'  1"  0"' 


Since  the  parts  of  a  foot  are  marked  by  accents,  the  fore- 
going principles  give  rise  to  the  following  law: 

Tfie  index  of  the  unit  of  any  product,  is  denoted  by  a  num- 
ber of  accents  equal  to  the  sum  of  the  indices  of  the  factors. 

1.   Multiply  6  ft.  7'  8"  by  2  ft.  9' 

ANALYSIS.  —  Multiply  by  9'.  Since 
8"=T£¥  ft,  and  9'=^  ft->  8"  x  9'=T¥?- 
x  T92  =  T^f  ,  ft,,  or  thirds.  Since  12 
thirds  make  1  second,  72//r=72-M2 
=  6"  ;  therefore  we  put  down  0'",  and 
carry  6".  T=  T7o,  and  7'  x  9'=  T75  x  T95 
=  J^  or  seconds  ;  63"+  6"=  69";  69" 
-r  12  =  5'  9"  ;  we  put  down  9"  and 

carry  5'.  6  ft.  x  9'=f  x  T\  =  f  f,  or  primes;  54'  +5'  =  59';  59' 
-M2  =  4  ft.  11';  which  we  set  down,  and  then  multiply  by  2  feet. 
8"  x  2  feet  =  16";  16"  H-  12  =  1'  4";  we  put  down  4"  under  the 
seconds,  and  carry  1":  7'  x  2  feet  =  14';  14'  +  1'  =  15';  15'  -f- 
12  =  1  ft.  3'  ;  we  put  3'  under  the  primes,  and  carry  I  ft.  :  6  ft. 
x  2  ft.  =  12  sq.  ft.  ;  12  ft.  +  1  ft.  =  13  ft.  ;  we  set  the  feet  under 
the  feet  and  add.  The  result  is  18  sq.  ft.  3'  1":  Hence,  the  fol- 
lowing 

Rule. 

I.  Write  the  multiplier  under  the  multiplicand,  so  that 
units  of  the  same  order  shall  fall  in  the  same  column. 

II.  Begin  with  the  lowest  unit  of  the  multiplier  and  the 
lowest   of  the   multiplicand,   and  make   the  index  of  each 
product  equal  to  the  sum  of  the  indices  of  the  factors. 

III.  Reduce  each  product,  in  succession,  to  square  feet, 
and  IWis  of  a  square  foot. 


APPLICATIONS.  201 

Examples. 

1.  Multiply  9  ft.  4  in.  by  8  ft.  3  in. 

2.  How  many  cords  and  cord  feet  in  a  pile  of  wood  24 
feet  long,  4  feet  wide,  and  3  'feet  6  inches  high  ? 

3.  Multiply  9  ft.  2  in.  by  9  ft.  6  in. 

4.  How  many  square  feet  are  there  in  a  board  17  feet  6 
inches  in  length,  and  1  foot  7  inches  in  width  ? 

5.  Multiply  24  feet  10  inches  by  6  feet  8  inches. 

6.  What  is  the  number  of  cubic  feet  in  a  granite  pillar 
3  feet  9  inches  in  width,  2  feet  3  inches  in  thickness,  and  12 
feet  6  inches  in  length  ? 

7.  Multiply  70  feet  9  inches  by  12  feet  3  inches. 

8.  There  is  a  certain  pile  of  wood,  measuring  24  feet  in 
length,  16  feet  9  inches  high,  and  12  feet  6  inches  in  width. 
How  many  cords  are  there  in  the  pile  ? 

9.  How  many  square  yards   in  the  walls   of  a  room,  14 
feet  8  inches  long,  1 1  feet  6  inches  wide,  and  7  feet  1 1  inches 
high? 

10.  If  a  load  of  wood  be  8  feet  long,  3  feet  9  inches  wide, 
and  6  feet  6  inches  high,  how  much  does  it  contain  ? 

11.  How   many  cubic   yards   of  earth   were   dug  from   a 
cellar  which  measured  42  feet  10  inches  long,  12  feet  6  inches 
wide,  and  8  feet  deep  ?     » 

12.  What  will  it  cost  to  plaster  a  room  20  feet  6'  long, 
16  feet  wide,  9  feet  6'  high,  at  18  cents  per  square  yard; 
and  making  allowance  for  a  door  that  is  6  feet .  6  in.  long, 
by  3  feet  3'  wide  ? 

13.  How  many  feet  of  boards,  1  inch  thick,  can  be  cut 
from  a  plank  18  feet  9  in.  long,  1  foot  6  in.  wide  and  3  in. 
thick,  if  there  is  no  waste  in  sawing? 

14.  What  will  be  the  cost  of  building  a  stone  wall,  45  ft. 
6  in.  long,   1  ft.  6'  thick,  and  37  ft.  9'  high,  at  $3.91  £  per 
cubic  yard  ? 

15.  How  many  loads  of  earth  must  be  taken  out  in  digging 
a  cellar  that  is  to  be  45  ft.  6  in.  long,  25  ft.  wide,  and  10  ft.  9 
in.  deep,  allowing  1  cubic  yard  of  earth  to  2  loads  ? 

9* 


202 


RATIO   AND   PROPORTION. 


RATIO   AND    PROPORTION. 

229.  A  RATIO  is  the  quotient   obtained   by  dividing  one 
number  by  another.  , 

230.  The  TERMS  of  a  ratio  are  the  divisor  and  dividend : 
hence,  every  ratio  has  two  terms. 

231.  The  divisor  is  called  the  ANTECEDENT. 

232.  The  dividend  is  called  the  CONSEQUENT. 

233.  The  ratio  of  one  number  to  another  is  expressed  in 
two  ways  : 

1st.   By  a  colon ;   thus,   3:12;   and  is  read,  3  is  to  12, 
or  12  divided  by  3  ; 

2d.    In  a  fractional  form ;   as,  ^.,   or  12  divided  by  3. 

234.  The  terms  of  a  ratio,  taken  together,   are  called  a 

COUPLET. 

235.  A  SIMPLE  RATIO  is  when  both  terms  of  a  couplet  are 
simple  numbers.     Thus,     6   :  18,     is  a  simple  ratio. 

236.  A  COMPOUND  RATIO,  is  one  which  arises  from  the  mul 
tiplication  of  two  simple  ratios :  thus,  in  the  simple  ratios, 

3   :  7,     and      6   :  8, 
if  we  multiply  the  corresponding  terms  together,  we  have 

3x6     :     7x8, 

which  is  compounded  of  the  ratio  of  3  to  7,  and  of  6  to  8. 
The  factors,  3  and  6,  are  called  ELEMENTS  of  the  first  term ; 
and  the  factors,  7  and  8,  are  ELEMENTS  of  the  second  term. 
The  elements  of  a  term  are  generally  written  in  a  column,  thus, 
3 
C 


:   g  I  ;  and  read,  3  multiplied  by  6,  to  7  multiplied  by  8. 


229.  What  is  a  ratio  ?— 230.  What  are  the  terms  of  a  ratio  ?   How 
many  terms  has  every  ratio?— 231.  What  is  the  divisor  called? 

232.  What  is  the  dividend  called? 

233.  In  how  many  ways  is  a  ratio  expressed?    What  are  they? 

234.  What  are  the  terms  of  a  ratio,  taken  together,  called? 

235.  What  is  a  simple  ratio?— 236.  What  is  a  compound  ratio? 


RATIO   AND   PROPORTION.  203 

237.  When  the  antecedent  is  less  than  the  consequent,  the 
ratio  shows  how  many  times  the  consequent  is  as  great  as 
the  antecedent. 

238.  When  the  antecedent  is  greater  than  the  consequent, 
the  ratio  shows  what  part  the  consequent  is  of  the  antecedent. 

NOTES. — 1.  Only  numbers  having  the  same  unit  value,  can  be 
compared  with  each  other:  hence,  all  numbers  compared,  mu-t 
be  reduced  to  the  same  unit. 

2.   The  ratio,  is  always  an  abstract  number. 

239.  To  measure  a  number,  is  to  find  how  many  times  it 
contains  another  number  of  the  same  kind,  which  is  called, 
the  standard.     The  unit  1,  is  the  simplest  standard  of  meas- 
ure, and  by  this,  all  numbers,  whether  integral  or  fractional, 
are  finally  measured. 

In  every  ratio,  the  antecedent  is  the  standard. 

Examples. 

1.  What  is  the  ratio  of  3  feet  to  6  feet? 

2.  What  is  the  ratio  of  10  dollars  to  40  dollars  ? 

3.  What  is  the  ratio  of  the  number  9  to  18  ? 

4.  What  is  the  compound  ratio  of  3  x  9  to  9x9? 

5.  What  is  the  compound  ratio  of  3x4  to  12x12? 

6.  What  is  the  compound  ratio  of  5x3x2  to  6x10x3? 


7.  What  part  of  9  is  2  ? 

8.  What  part  of  16  is  4? 

9.  What  part  of  100  is  20  ? 

10.  What  part  of  300  is  200  ? 

11.  What  part  of  144  is  36  ? 


12.  3  is  what  part  of  12  ? 

13.  5  is  what  part  of  20? 

14.  8  is  what  part  of  56  ? 

15.  7  is  what  part  of  8  ? 

16.  12  is  what  part  of  132? 


237.  What  does  the  ratio  show,  when  the  antecedent  is  less  than 
the  consequent  ? 

238.  What   does    the  ratio    show,  when  the  antecedent  is  the 
greater  ? 

239.  What  is  the  operation  of  measuring  a  number  ?    What  is 
the  measure  called  ?    What  is  the  simplest  standard  for  all  num- 
bers ?    What  is  the  standard  in  any  ratio  ? 


204  RATIO   AND    PROPORTION. 

NOTE. — The  standard  is  generally  preceded  by  the  word  of, 
and  in  comparing  numbers,  may  be  named  second,  as  in  exam- 
ples 12,  13,  14,  15,  and  16;  but  it  must  always  be  used  as  a 
divisor,  and  should  be  placed  first  in  the  statement. 


17.  What  part  of  f  is  J  ? 

18.  i  of  f  is  what  part  of  T9T 


19.  4J  is  what  part  of  9£ 

20.  f  is  what  part  of  4J  ? 


21.  2.15  is  what  part  of  6.975? 

22.  5|  is  what  part  of  7.1875? 

23.  What  is  the  ratio  of  2  T.  3  cwt.  2  qr.  to  1  T.  11  cwt. 
3qr.  16  lb.? 

24.  What  is  the  ratio  of  1  mi.  6  fur.  8  rd.  to  10  mi.  1  fur. 
16  rd.  1  yd.  2ft,? 

25.  The  ratio  of  two  numbers  is  3,  and  the  antecedent 
16 :  what  is  the  consequent  ? 

ANALYSIS. — Since  the  ratio  is  equal  to  Eatio  consequent 
the  consequent  divided  by  the  antecedent,  "  antecedent* 

it  follows, 

1st.  That  the  consequent  is  equal  to  the  antecedent  multiplied 
J>y  the  ratio: 

2d.  That  the  antecedent  is  equal  to  the  consequent  divided  by 
the  ratio. 

26.  The  ratio  of  two  jmnibers  is   6,  and  the  antecedent 
12  :  what  is  the  consequent  ? 

27.  The  ratio  of  two  numbers  is  9,  and  the  consequent 
108  :   what  is  the  antecedent  ? 

28.  The  ratio  of  two  numbers  is  5,  and  the   consequent 
125  :   what  is  the  antecedent  ? 

29.  .The  ratio  of  two  numbers  is  f,  and  the  antecedent  ^ : 
what  is  the  consequent  ? 

30.  The  ratio  of  two  numbers  is  •§-,  and  the  consequent  f : 
what  is  the  antecedent  ? 

31.  The  ratio  of  two  numbers  is  6,  and  the   consequent 
12  :   what  is  the  antecedent  ? 

32.  The  antecedent  is  i,  and  the  consequent  J  :   what  is 
the  ratio  ? 

33.  The    antecedent    is    3x6x9,    and    the    consequent 
1x5x4x2:  what  is  the  ratio  ? 


RATIO   AND    PROPORTION.  205 


SIMPLE    PROPORTION. 

240.  A  SIMPLE  PROPORTION  is  the  comparison  of  the  terms 
of  two  equal  simple  ratios. 

Thus,  the  ratio  of  3  :  6,  is  2  ;  and  the  ratio  of  8  :  16,  is 

2  ;   and  we  compare  the  terms  by  writing  a  double  colon  be- 
tween the  couplets  ;   thus, 

3     :     6     ::     8     :     16; 
which  is  read,       3  is  to  6,  as   8    to    16. 

Hence,  every  proportion  has  two  couplets  and  four  terms. 

NOTE. — When  the  ratio  of  the  first  couplet  is  greater  than  1, 
the  second  term  is  greater  than  the  first,  and  the  fourth  term 
greater  than  the  third.  When  the  ratio  is  less  than  1,  the  second 
term  is  less  than  the  first,  and  the  fourth  term  less  than  the 
third. 

241.  The  first  and  fourth  terms  of  a  proportion  are  called 
the  extremes :  the  second  and  third  terms,  the  means.    Thus, 
in  the  proportion, 

3     :     12     :  :     6     :     24, 

3  and  24  are  the  extremes,  and  12  and  6  the  means. 

242.  Since  the  ratio  in  the  first  couplet  is  equal  to  that 
in  the  second,  we  have, 

12  =  24 
S  "  6  ' 
and  we  shall  have,  by  reducing  to  a  common  denominator, 

12  X  6  _  24  x  3 
3x6         6x3' 

240.  What  is  a  Simple  Proportion?    How  is  it  written? 
NOTE. — When  the  ratio  in  the  first  couplet  is  greater  than  1, 

what  follows  ? 

241.  What  are  the  first  and  fourth  terms  of  a  proportion  called? 
The  second  and  third? 

242.  What  is  the  product  of  the  extremes  equal  to  ? 


206 


RATIO   AND    PROPORTION. 


Since  the  fractions  are  equal,  and  have  the  same  denomi- 
nators, their  numerators  must  be  equal,  viz. : 

12  x  6  =  24  X  3  ;   that  is, 

In  any  proportion,  the  product  of  the  extremes  is  equal 
to  the  product  of  the  means. 

243.  Since,  in  any  proportion,  the  product  of  the  extremes 
is  equal  to  the  product  of  the  means,  it  follows  that, 

1st.  Either  extreme  is  equal  to  the  product  of  the  means 
divided  by  the  other  extreme. 

2d.  Either  mean  is  equal  to  the  product  of  the  extremes 
divided  by  the  other  mean. 

NOTE. — We  shall  denote  the  required  term  of  a  proportion  by 
the  letter  x. 

Examples. 
1.    In  the  proportion, 

6     :     12     :     24     :     ar, 
find  the  value  of  the  fourth  term : 

„   x  ==  12x24  ==  4g< 


Find  the  value  of  the  required  term,  in  the  following  pro- 
portions : 

x 
10 
54 

$3 


45. 
50. 
30. 
4  yards. 


9     :      x      :  :     15 

8  :     10     :  :      x 

9  :  :  :      x 
$15     :     $3     :  :      x 

i     :     i        ::      x      :     TV 

To  what  number  has  5  the  same   ratio   as  exists  be- 
tween 2  and  4  ? 

8.  To  what  number  has  one-half  the  same  ratio  as  exists 
between  3  and  21  ? 

9.  To  what  number  has  5  the  same   ratio   as   exists  be- 
tween 6  and  18? 

243.  What  is  either  extreme  equal  to  ?    Either  mean  ?    By  what 
is  the  required  term  of  a  proportion  designated  ? 


RATIO   AND    PROPORTION  207 


COMPOUND    PROPORTION. 

244.   A  COMPOUND  PROPORTION  is   the   comparison   of  the 
terms  of  two  equal  ratios,  when  one  or  both  are  compound : 


thus,  :  ::      10      :     20; 

Or  2i          3l  51          15 

Ur'  6  j      '     8  j      *  '      6 }     '       4 


Any  compound  proportion  may  be  reduced  to  a  simple  one, 
by  multiplying  the  elements  of  each  term  together  :  thus,  by 
multiplying  the  elements  in  the  last  proportion,  we  have 
12     :     24       :  :       30     :     60. 

Hence,  in  any  compound  proportion, 

The  product  of  the  extremes  is  equal  to  the  product  of 
the  means:  therefore,  in  a  compound  proportion,  we  can  find 
the  required  term,  as  in  Art.  243. 

What  are  the  required  terms  in  the  following  proportions : 

1.  5  X  10   :     3  X  7     :  :    48   :  x. 

2.  2  x     4   :  16  x  6     :  :      9   :  x. 

If  all  the  parts  of  a  compound  proportion  are  known,  ex- 
cept one  element,  as  in  the  proportion 

2)          3)  5)  15 

6J          8[  6f          x 

that  element  is  equal  to  the  product  of  the  means  divided 
by  the  product  of  the  elements  of  the  first  term  and  the 
knoion  elements  of  the  fourth  term :  thus, 
3x8x5x6  _ 
2  x  6  x  15 

3.  What  is  the  required  element  in  the  proportion, 

3 


)          5)  3)          2) 

[     :     d      ::      d     :     5J 

)          8)  9)          x) 


244.  What  is  a  compound  proportion  ? 


208 


SINGLE    RULE   OF   THREE. 


OPERATION. 

3  :  6  : :  $12  :  x. 
C  =  ^i-2  =  $24. 


RULE    OF    THREE. 

245.  The  RULE  OF  THREE  is  the  process  of  finding,  from 
three  given  numbers,  a  fourth,  to  which  one  of  the  given 
numbers  shall  have  the  same  ratio  as  exists  between  the 
other  two. 

The  Single  Rule  of  Three  embraces  all  the  cases  of  Simple 
Ratios. 

1.  If  3  yards  of  cloth  cost  $12,  what  will  6  yards  cost? 
ANALYSIS. — The    quantity,    3    yards, 

bears  the  same  ratio  to  the  quantity, 
6  yards,  as  $12,  the  cost  of  3  yards,  to 
x  dollars,  the  cost  of  6  yards :  the  4th 
term  is  found  by  multiplying  the  sec- 
ond and  third  terms  together,  and  di- 
viding the  product  by  the  first  (Art. 
243). 

2.  If  6  barrels  of  flour  will  last  a  family  of  5  persons,  8 
months,  how  long  will  they  last  a  family  of  10  persons  ? 

ANALYSIS. — Write  the  required  term 
(a?),  in  the  4th  place,  and  the  term  8, 
having  the  same  unit  value,  in  the  third 
place :  then  consider  whether  the  third 
term  is  greater,  or  less,  than  the  4th:  10 

wrhen  greater,  place  the  greater  of  the 

remaining  terms  in  the  first  place,  and  when  less,  place  the  less 
term  there:  then  find  the  value  of  the  4th  term. 

In  this  example,  it  is  plain,  that  the  same  provisions  (6  barrels 
of  flour)  will  not  last  ten  persons  as  long  as  it  did  5;  hence,  the 
3d  term  will  be  greater  than  the  4th,  which  requires  the  first 
term  to  be  greater  than  the  2d;  hence,  we  write  10  in  the  first 
place,  and  5  in  the  second. 

In  the  first  example,  it  is  plain,  that  the  3d  term,  $12,  will  be 
less  than  the  4th;  therefore,  the  less  of  the  remaining  terms, 
3,  is  written  in  the  first  place.  Hence,  the  following 


OPERATION. 

It)  :  5  : :  S,  :  x. 

5x8 


245.  What  is  the  Rule  of  Three?    What  is  the  Single  Rule  of 
Three?    Give  the  rule. 


SINGLE   RULE   OF  THREE.  209 

Rule. 

1.  Write  the  required  term  '(a?)  in  the  4th  place,  and  the 
term  having  the  same  unit  value   in  the   3d  place :    (h<'n 
consider,  from  the  nature  of  the  question,  whether  the  4th 
term  is  greater  or  less,  than  the  3cZ:   when  greater,  p'ace 
the  less  of  the  remaining  terms  in  the  first  'place,  and  when 
it  is  less,  place  the  greater  term  there,  and  the  remaining 
term  in  the  second  place. 

II.    Then  multiply  the  second  and  third  terms  together, 
and  divide  their  product  by  the  first. 

NOTES. — 1.   If  the  first  aud  second  terms  have  different  units, 
they  must  be  reduced  to  the  same  unit. 

2.  If  the  third   term  is   a   compound  denominate  number,   it 
must  be  reduced  to  its  smallest  unit. 

3.  The  preparation  of  the  terms,  and  writing  them  in  their 
proper  places,  is  called,  the  Statement. 

Examples. 

1.  If  I  can  walk  84  miles  in  3  days,  how  far  can  I  walk 
in  1 1  days  ? 

2.  If  4  hats  cost  $12,  what  will  be  the  cost  of  55  hats, 
at  the  same  rate  ? 

3.  If  a  certain  quantity  of  food  will  subsist  a  family  of  12 
persons,    48    days,   how  long  will   the   same   food  subsist   a 
family  of  8  persons  ? 

4.  If  40  yards   of  cloth  cost  $170,  what  will  325  yards 
cost,  at  the  same  rate  ? 

5.  If  240  sheep  produce   660  pounds  of  wool,  how  many 
pounds  will  be  obtained  from  1200  sheep? 

6.  If  30  barrels  of  flour  will  subsist  100  men  for  40  days, 
how  long  will  it  subsist  25  men  ? 

7.  If  2  gallons   of  molasses   cost  65   cents,   what  will  3 
hogsheads  cost  ? 

8.  If  a  man  travels  at  the  rate  of  210  miles  in  6  days, 
how  far  will  he  travel  in  a  year,  supposing  him  not  to  travel 
on  Sundays  ? 


210 


SINGLE    RULE    OF   THREE. 


9.  If  90  bushels  of  oats  will  feed  40  horses  for  six  days, 
how  many  horses  would  consume  the  same  in  12  days  ? 

10.  If  4  yards  of  cloth  cost  $13,  what  wiU  be  the  cost 
of  3  pieces,  each  containing  25  yards  ? 

11.  If  48  yards  of  cloth  cost  $67.25,  what  will  144  yards 
cost,  at  the  same  rate? 

12.  If  3  common   steps,  or  paces,  are  equal  to   2  yards 
how  many  yards  are  there  in  160  paces  ? 

13.  If   750   men   require    22500   rations    of  bread    for   a 
month,  how  many  rations  will  a  garrison  of  1200  men  require? 

14.  A  certain  work  can  be  done  in  12  days,  by  working 
4  hours  a  day:    how  many  days  would  it  require  the  same 
number  of  men  to  do  the  same  work,  if  they  worked  6  hours 
a  day? 

15.  A  pasture  of  a  certain  extent,  supplies  30  horses  for 
18  days:  how  long  will  the  same  pasture  supply  20  horses? 

16.  If  14i  yards  of  cloth  cost  $19|,  how  much  will  19f- 
yards  cost  ? 

yds.         yds.  $ 

NOTE.— Make  the   statement,   and        14i  :  19-J-  :  /£. 

then  change  the  mixed  to  improper 
fractions ;  and  afterwards,  multiply 
the  2d  and  3d  terms  together,  and 
divide  by  the  1st. 


191  = 


2~- 


39 


159    _    6201    . 

1  6       > 


Xi  o  a    
~~Q             


6201 


29    

2 


6201 

~w 


v  _  —   6201   _ 

X  29  -   2^"  ' 


Or,      ¥xIi-¥  =        Xx       = 


17.  If  2  Ib.  of  beef  cost  |  of  a  dollar,  what  will  30  Ib. 
cost  ? 

18.  If  6  men  can  dig  a  ditch  in  40  days,  what  time  will 
30  men  require  to  dig  the  same  ? 

19.  If  1T4T  bushels  of  wheat  cost  $2f,  how  much  will  60 
bushels  cost  ? 


APPLICATIONS.  211 

20.  If  4|  yd.  of  cloth  cost  $9.75,  what  will  13£  yd.  cost? 

21.  If  f  of  a  yard  of  cloth  costs  \  of  a  dollar,  what  will 
2J  yards  cost  ? 

22.  If  >%  of  a  ship  costs  £273  2s.  6d.,  what  will  35j  of 
her  cost  ? 

23.  If  a  post,  8  feet  high,  casts  a  shadow  12  feet  in  IcnjrMi, 
what  must  be  the  height  of  a  tree  that  casts  a  shadow  1^2 
feet  in  length,  at  the  same  time  of  day  ? 

24.  If  a  man  performs  a  journey  in   22^  days,  when  the 
days  are  12  hours  long,  how  many  days  will  it  take  him  to 
perform  the  same  journey,  when  the  days  are  15  hours  long? 

25.  If  7  cwt.  1  qr.  of  sugar  cost  $64.96,  what  will  be  the 
cost  of  4  cwt.  2  qr.  ? 

26.  A  merchant,  failing  in  trade,  pays  65  cents  for  every 
dollar  which  he  owes  ;    he   owes  A  $2750,  and  B  $1975  : 
how  much  does  he  pay  each  ? 

27.  If  a  person  drinks  20  bottles  of  wine  per  month,  when 
it  costs  2s.  per  bottle,  how  much  must  he  drink  without  in- 
creasing the  expense,  when  it  costs  2s.  6d.  per  bottle  ? 

28.  If  6  sheep  cost  $15,  and  a  lamb  costs   one-third  as 
much  as  a  sheep,  what  will  27  lambs  cost  ? 

29.  If  4^-  gallons  of  molasses  cost  $2|,  how  much  is  it 
per  quart  ? 

30.  A  man  receives  f  of  his  income,  and  finds  it  equal  to 
$3724.16 :  how  much  is  his  whole  income  ? 

31.  A  cistern,  containing  200  gallons,  is  filled  by  a  pipe 
which  discharges  3  gallons  in  5  minutes ;  but  the  cistern  has 
a  leak,  which  empties  at  the  rate  of  1  gallon  in  5  minutes : 
if  the  water  begins  to  run  in  when  the  cistern  is  empty,  how 
long  will  it  run  before  filling  the  cistern  ? 

32.  If  9  men,  in  18  days,  will  cut  150  acres  of  grass,  how 
many  men  will  cut  the  same  in  27  days  ? 

33.  If  a  garrison   of  536   men   have   provisions   for  326 
days,  how  long  will  those  provisions  last,  if  the  garrison  be 
increased  to  1304  men  ? 

34.  If  4   barrels   of  flour  cost  $34f,   how  much   can   be 
bought  for 


212 


SINGLE    KULK    OF   THREE. 


35.  If  2|  gallons  of  molasses   cost  65  cents,  what  will  3| 
hogsheads  cost  ? 

36.  What  is  the  cost  of  6  bushels  of  coal,  at  the  rate  of 
£1  14s.  6d.  a  chaldron? 

37.  What  quantity  of  corn  can  I  buy  for  90  guineas,  at 
the  rate  of  5  shillings  a  bushel  ? 

38.  A   merchant,  failing  in  trade,    owes   $3500,    and   his 
effects   arc   sold   for  $2100  :    how  much  does  B  receive,   to 
whom  he  owes  $420  ? 

39.  If  3  yards  of  broadcloth  cost  as  much  as  4  yards  of 
cassimere,  how  much  cassimere  can  be  bought  for  18  yards 
of  broadcloth  ? 

40.  What  length  must  be  cut  off  from  a  board  that  is  9 
inches  wide,  to  make  a  square  foot,  that  is,  as  much  as  is 
contained  in  12  inches  in  length  and  12  in  breadth  ? 

41.  If  7  hats  cost  as  much  as  25  pair  of  gloves,  worth 
84  cents  a  pair,  how  many  hats  can  be  purchased  for  $216  ? 

42.  How  many  barrels  of  apples  can  be  bought  for  $114.33, 
if  7  barrels  cost  $21.63? 

43.  If  27  pounds  of  butter  will  buy  15  pounds  of  sugar, 
how  much  butter  will  buy  36  pounds  of  sugar? 

44.  If  42J  tons  of  coal  cost  $206.21,  what  will  be  the 
cost  of  2|  tons  ? 

45.  If  a  certain  sum  of  money  will  buy  40  bushels  of  oats, 
at  45  cents  a  bushel,  how  many  bushels  of  barley  will  the 
same  money  buy,  at  72  cents  a  bushel  ? 

46.  If  40  gallons  run  into  a  cistern,  holding  700  gallons, 
in  an  hour,  and  15  run  out,  in  what  time  will  it  be  filled? 

47.  A  piece  of  land  of  a  certain  length,  and  12|  rods  in 
width,  contains  1J  acres :  how  much  would  there  be  in  a  piece 
of  the  same  length,  26§  rods  wide  ? 

48.  If  13  men  can  be  boarded  1  week  for  $39.585,  what 
will  it  cost  to  board  3  men  and  6  women  the  same  time,  the 
women  being  boarded  at  half  price  ? 

49.  What   will   75   bushels    of  wheat   cost,   if  4   bushels 
3  pecks  cost  $10.687  ? 


DOUBLE   RULE   OF   THREE.  213 


DOUBLE  RULE   OF  THREE. 

246.  The  DOUBLE  RULE  OF  THREE  is  an  application  of  the 
principles  of  Compound  Proportion. 

1.  If  a  family  of  6  persons  expend  $300  in  8  months,  how 
much  will  serve  a  family  of  15  persons  for  20  months  ? 

ANALYSIS.— Write    the    re-  STATEMENT. 

quired  term  in  the  4th  place,  g  |          15  } 

and  $300,  having   the    same  g  }    :    20 }     :  :               :  x' 
unit,  in  the  3d  place.     Write 

the    elements    of    the    term  x  _  15  X  20  X  300 

named    in    connection    with  6x8 
$300,   in  the   1st  place,   and 

the  elements  of  the  remaining  term  in  the  2d  place:   then  find 
the  value  of  a-,  as  in  Art.  243. 

2.  If  32  men  build  a  wall  36  feet  long,  8  feet  high,  and 
4  feet  thick,  in  4  days,  working  12  hours  a  day,  how  long  a 
wall,  that  is  6  feet  high  and  3  feet  thick,  can  48  men  build 
in  36  days,  working  9  hours  a  day? 

ANALYSIS. — In  this  example,  each  term  of  the  proportion  con- 
tains 3  elements,  and  the  required  element  is  the  length  of  the 
second  wall.  Denote  this  element  by  a-,  and  write  it,  and  the 
other  elements  of  the  term,  6  and  3,  in  the  4th  place,  as  in  the 
statement  below.  Then,  write  the  term  whose  elements  have  the 
same  units,  in  the  3d  place ;  the  term  mentioned  in  the  question 
in  connection  with  the  third  term,  in  the  first  place ;  and  the  re- 
maining term  in  the  second  place :  this  will  give, 


STATEMENT. 


32)  48 

4V       :      36 

12  9 


I  V  "} 


This  arrangement  gives, 

labor    :    labor    :  :    work  done    :    work  done. 

246.  What  is  tlie  Double  Rule  of  Three  ?    Give  the  Rule. 


214  DOUBLE   KULE   OF   THREE. 

When  quantity  and  cost  are  considered,  it  will  give, 

quantity  :  quantity     :  :     cost  :  cost. 
Finding  the  required  element  (Art.  244),  we  have, 
48X36X9X86X8X4 


32x4x12x6x3 


Hence,  we  have  the  following 
Rule. 

Write  the  term  which  contains  the  required  element  in 
the  fourth  place;  the  term  having  like  units,  in  the  third 
place;  the  term  mentioned  in  connection  with  the  third 
term,  in  the  first  place  ;  and  the  remaining  term  in  the 
second  place.  Then  find  the  value  of  the  required  element, 
as  in  Art.  244. 

Examples.  , 

1.  If  I  pay  $24  for  the  transportation  of  96  barrels  of 
flour  200  miles,  what  must  I  pay  for  the  transportation  of 
480  barrels  75  miles  ? 

2.  If  12  ounces  of  wool  be  sufficient  to  make  1|  yards  of 
cloth  6  quarters  wide,  what  number  of  pounds  will  be  required 
to  make  450  yards  of  flannel  4  quarters  wide  ? 

3.  What  will  be  the  wages  /of  9  men  for  11  days,  if  the 
wages  of  6  men  for  14  days  be  $84? 

4.  How  long  would  406  bushels  of  oats  last  7  horses,  if 
154  bushels  serve  14  horses  44  days  ? 

5.  If  a  man  travels  217  miles  in  7  days,  traveling  6  hours 
a  day,  how  far  would  he  travel  in  9  days,  if  he  traveled  11 
hours  a  day? 

6.  How  long  will  it  take  5  men  to  earn  $11250,  if  25 
men  can  earn  $6250  in  2  years  ? 

7.  If  15  weavers,  by  working  10  hours  a  day  for  10  days, 
^can  make  250  yards  of  cloth,  how  many  must  work  9  hours 

a  day  for  15  days,  to  make  60  7  £  yards  ? 


APPLICATIONS.  215 

8.  A  regiment  of  100  men  drank  20  dollars'  worth  of  wine, 
at  30  cents  a  bottle :  how  many  men,  having  the  same  allow- 
ance, will  require  12  dollars'  worth,  at  25  cents  a  bottle  ? 

9.  If  a  footman  travels  341   miles  in  7^   days,  traveling 
12|  hours  each  day,  in  how  many  days,  traveling  10J  hours 
a  day,  will  he  travel  155  miles? 

10.  If  25  persons  consume  300  bushels  of  corn  in  1  year, 
how  much  will   139   persons   consume  in   8   months,    at   the 
same  rate  ? 

11.  How  much  hay  will  32  horses  eat  in  120  days,  if  96 
horses  eat  3|  tons  in  7£  weeks  ? 

12.  If  $2.45  will  pay  for  painting  a  surface  21  feet  long 
and  13 J  feet  wide,  what  length  of  surface  that  is  lOf  feet 
wide,  can  be  painted  for  $31.72? 

13.  How  many  pounds  of  thread  will  it  require  to  make 
60  yards  of  3  quarters  wide,  if  7  pounds  make  14  yards  6 
quarters  wide  ? 

14.  If  500  copies  of  a  book,  containing  210  pages,  require 
12  reams  of  paper,  how  much  paper  will  be  required  to  print 
1200  copies  of  a  book  of  280  pages  ? 

15.  If  the  transportation  of  9  T.  15  cwt.  20  Ib.  for  260 
miles,   costs   $76.50,  what  will  be   the   cost   of  transporting 
25  T.  16  cwt.  3'qr.  for  189  miles,  at  the  same  rate? 

16.  If  a  cistern,  17J  feet  long,  10^  feet  wide,  and  13  feet 
deep,  holds   546   barrels   of  water,  how  many  barrels  will  a 
cistern  12  feet  long,  10  feet  wide,  and  7  feet  deep,  contain  ? 

17.  A  contractor  agreed  to  build  24  miles  of  railroad  in 

8  months,  and  for  this  purpose,  employed  150  men  ;   at  the 
end  of  5  months,  but  10  miles  of  the  road  were  built :  how 
many  more  men  must  be  employed,  to  finish  the  road  in  the 
time  agreed  upon  ? 

18.  If  336   men,  in  5  days  of  10  hoars   each,  can  dig  a 
trench  of  5  degrees  of  hardness,  70  yards  long,  3  wide,  and 
2  deep :  what  length  of  trench  of  6  degrees  of  hardness,  5 
yards  wide  and  3  yards  deep,  ma^  be  dug  by  240  men,  in 

9  days  of  1 2  hours  each  ? 


216  PARTNERSHIP. 


PARTNERSHIP. 

247.  A   PARTNERSHIP   is   an   association   of  two   or  more 
persons,  under  an  agreement  to  share  the  profits  and  losses  of 
business.     The  persons  thus  associated,  are  called,  Partners. 

248.  CAPITAL,  or  STOCK,  is  the  amount  of  money  or  prop- 
erty contributed  by  the  partners,  and  used  in  the  business. 

249.  DIVIDEND  is  the  gain  or  profit,  divided  to  each  partner. 

250.  Loss,  is  the  opposite  of  Gain  or  Profit. 

251.  When   the   capital  of  each  partner  is  employed  for 
the  same  time. 

Since  the  Capital  or  Stock  produces  the  gain  or  profit, 
each  man's  share  should  be  proportional  to  his  amount  of 
Stock  :  Hence,  we  have, 

Whole  Stock  :   each  man's  Stock  ::  Whole  Profit  :  each  man's  Profit. 

. "** 

Examples. 

1.  A  and  B  buy  certain  goods,  amounting  to  $160,  of 
which  A  pays  $90,  and  B  $70  ;  they  gain  $32,  by  the  sale 
of  them :  what  is  the  share  of  each  ? 

OPERATION. 

9         2 

160    :     90    ::     32     :     x  =  ^0  *  ^  =  $18,  A's  share. 


7        2 
160    :     70    : :     32    :     x  -  -    ^^  =  $14,  B's  share. 


247.  What  is  a  Partnership?    What  are  partners? 

248.  What  is  capital,  or  stock  ?— 249.  What  is  dividend  ? 

250.  What  is  loss  ?  * 

251.  What  produces  the  gain  or  profit  ?    What  is  the  rule  for 
finding  each  man's  share? 


PARTNERSHIP.  217 

2.  A  and  B  have  a  joint  stock  of  $2100,  of  which  A  owns 
$1800   and  B  $300;   they  gain  in   a  year,  $1000:    what  is 
each  one's  share  of  the  profits  ? 

3.  A,  B,  and   C   fit   out   a   ship   for   Liverpool.     A   con- 
tributes $3200,   B  $5000,  and  C  $4500 ;    the  profits   of  the 
voyage  amount  to  $1905:   what  is  the  portion  of  each  '( 

4.  A,  B,  and  C  agree  to  build  a  railroad,  and  contribute 
$18000  of  capital,  of  which  B  pays  2  dollars,  and  C  3  dol- 
lars, as  often  as  A  pays  1  dollar;   they  lose  $2400  by  tin* 
operation  :   what  is  the  loss  of  each  ? 

5.  Three  drovers  hire  a  pasture  for  6  weeks,  at  an  expense 
of  $275;   the  first  puts  in  300  cattle,  the  second  450,  and 
the  third  500 :   what  ought  each  to  pay  ? 

6.  Two   merchants   enter  into   partnership.     One   puts   in 
$5000,  and  the  other  $2000.     The  partner  that  put  in  the 
less  sum,  is  to  receive  $300  extra  for  his  superior  knowledge 
of  the  business.    They  gain  $4725  :  what  is  the  share  of  each  9 

7.  A,  B,  and  C  make  up  a  capital  of  $20000 ;  B  and  C 
each  contribute  twice   as  much  as  A ;    but  A  is  to  receive 
one-third  of  the  profits  for  extra  services ;  at  the  end  of  the 
year,  they  have  gained  $4000 :   what  is  each  to  receive  ? 

8.  Three  merchants  own  a  ship,  in  the  following  propor- 
tions :  -5-,  i,  i-     The  ship  required  repairs,  to  the  amount  of 
$1350  :   what  was  each  one's  share  of  the  expense  ? 

252.    When  the  capital  is  employed  for  unequal  times. 

When  the  partners  employ  their  capital  for  unequal  periods 
of  time,  the  profit  of  each  will  depend  on  the  two  elements, 
Capital  and  Time,  and  will  be  proportional  to  their  product : 
Hence, 

Multiply  each  man's  stock  by  the  time  he  continued  it  in 
trade:  then  say, 

.    As  the  sum  of  the  products    :    the  whole  gain  or  loss, 
V  '  ::    eacb  product    :    each,  man's  share. 

253.  What  arc  the  elements  of  profit,  when  the  capital  is  employ 
for  nnequaTttllMM  ?    WIlaTis  the  rule  for  finding  the  profit  of 
partner? 

10 


218 


PARTNERSHIP. 


Examples. 

1.  A  and  B  entered  into  partnership.  A  put  in  $840 
for  4  months,  and  B,  $650  for  6  months  ;  they  gained  $363 : 
what  is  each  one's  share  ? 


OPERATION. 


A. 
B. 


x  4  =  3360 
650  x  6  =  3900 

7260 


363 


3360 
3900 


$168,  A's.  ) 
$195,  B's.  [ 


2.  A  puts  in  trade   $550  for  7  months,   and  B  pnts  in 
$1625  for  8  months;  they  make  a  profit  of  $337:  what  is 
the  share  of  each  ? 

3.  A  and  B  hire  a  pasture,  for  which  they  agree  to  pay 
$92.50;  A  pastures  12  horses  for  9  weeks,  and  B,  11  horses 
for  7  weeks :   what  portion  must  each  pay  ? 

4.  Four  traders  form  a  company.     A  puts  in  $400  for  5 
months ;  B,  $600  for  7  months ;   C,  $960  for  8  months ;  D, 
$1200  for  9  months.     In  the  course  of  trade,  they  lost  $750 : 
how  much  falls  to  the  share  of  each  ? 

5  A,  B,  C  contribute  to  a  capital  of  $15000,  in  the  fol- 
lowing manner :  every  time  A  puts  in  3  dollars,  B  puts  in 
$5,  and  C  $7.  A's  capital  remains  in  trade  1  year,  B's  1-f 
years,  and  C's  2f  years  ;  at  the  end  of  the  time,  there  is  a 
profit  of  $15000  :  what  is  the  share  of  each  ? 

6.  A  commenced  business  January  1st,  with  a  capital  of 
$3400.     April  1st,  he  took  B  into  partnership,  with  a  capi- 
tal of  $2600 ;  at  the  expiration  of  the  year,  they  had  gained 
$750 :  what  is  each  one's  share  of  the  gain  ? 

7.  'James  Fuller,  John  Brown,  and  William  Dexter  formed 
a  partnership,  under  the  firm  of  Fuller,  Brown  &  Co.,  with 
a    capital    of   $20000  ;    of   which    Fuller   furnished    $6000, 
Brown  $5000,  and  Dexter  $9000.     At  the  expiration  of  4 
months,  Fuller  furnished  $2000  more ;    at  the  expiration  of 
6  months,  Brown  furnished  $2500  more ;   and  at  the  end  of 
a  year,  Dexter  withdrew  $2000.     At  the  expiration  of  one 
year  and  a  half,  they  found  their  profits  amounted  to  $5400  : 
what  was  each  partner's  share  ? 


PERCENTAGE.  219 

PERCENTAGE. 

253.  PER  CENT,  means,  by  the  hundred.     Thus,  1  per  cent, 
of  a  number,  is  one-hundredth  of  it ;    2  per  cent.,  two-hun- 
dredths  ;  3  per  cent.,  3  hundredths,  &c. 

254.  The   RATE   PER  CENT,  is   the   number  of  hundredths 
taken ;  thus,  if  1  hundredth  is  taken,  the  rate  is  1  per  cent. ; 
if  2  hundredths,  the  rate  is   2  per  cent. ;   if  3  hundredths, 
3  per  cent.,  &c. 

255.  The  BASE  of  porrcnt:^'',  is  the  number  on  which  the 
percentage  is  colllpllted  ;  and   the  result  of  the  computation 

256.  The  rate  per  cent,  is  generally  expressed  decimally ; 
thus, 

1  per  cent,  of  a  number,  is  T^  of  it  =  .01       of  it. 

3  per  cent,  of  a  number,  is  Tg^  of  it  =  .03       of  it. 

A  per  cent,  of  a  number,  is  ffa  of  it  ==  .5  of  it. 

per  cent,  of  a  number,  is  }%%  of  it  =  1  time  it. 

16  per  cent,  of  a  number,  is  JJ-J-  of  it  =  1.16  of  it. 

00  per  cent,  of  a  number,  is  f g-g-  of  it  =  2  times  it. 

i  per  cent,  of  a  number,  is  Tg^  of  it  =  .005     of  it. 

j  per  cent,  of  a  number,  is  Tf^  of  it  =  .0075  of  it. 

.7  per  cent,  of  a  number,  is  T'^¥  of  it  =  .007  of  it. 

.45  per  cent,  of  a  number,  is  j45^  of  it  =  .0045  of  it. 

.5J  per  cent,  of  a  number,  is  -j^  of  it  =  .0055  of  rt. 

Write,  decimally,  2  per  cent. ;  8J  per  cent. ;  6 J  per  cent. ; 
f  per  cent.;  |  per  cent.;  117  per  cent.;  205  per  cent.;  .9 
per  cent. ;  275.25  per  cent. 

253.  What  is  the  meaning  of  per  cent.  ? 

254.  What  is  the  rate  per  cent.? 

255.  What  is  the  base  of  percentage  ?    What  is  Percentage  ? 

256.  How  is  the  rate  per  cent,  generally  expressed? 


3 

50 
100 
116 
200 


220 


PERCENTAGE. 


257.    Having   given   the   base   and    rate,   to   find   the   per- 
centage. 

1.  What  is  the  percentage  of  $320,  the  rate  being  5  per 
cent.  ? 

ANALYSIS. — The  base  is  $320,  and  the  rate 
being  5  per  cent.,  is  expressed  decimally  by 
.05.  We  are  then  to  take  .05  of  the  base; 
this  we  do,  by  multiplying  $320  by  .05. 

Hence,  to  find  the  percentage  of  a  number, 


OPERATION. 

320 
.05 


$16.00,  Ans. 


Rule. — Multiply  the  number  by  the  rate,  expressed  deci- 
mally, and  the  product  will  be  the  percentage. 


Examples. 

1.  What  is  the  percentage  of  $657,  the  rate  being  4J  per 
cent.  ?  %  V-  V 

OPERATION. 

NOTE. — When  the  rate  cannot  be  Xr* 

reduced  to  an   exact  decimal,  it  is  -v.j 

most  convenient  to  multiply  by  the 
fraction,  and  then  by  that  part  of 
the  rate  which  is  expressed  in  exact 
decimals. 


219  =  J  per  cent. 
2628  =  4  per  cent. 

$28.47  =  4J  per  cent. 


Find  the  percentage  of  the  following  numbers : 


2.  2J  per  cent,  of  650  dollars. 

3.  3  per  cent,  of  650  yards. 

4.  4^  per  cent,  of  875  cwt. 

5.  6|  per  cent,  of  $37.50. 

6.  '5 1  per  cent,  of  2704  miles. 

7.  |  per  cent,  of  1000  oxen. 

8.  2f  per  cent,  of  $376. 

9.  5|  per  cent,  of  $327.33.       17.   .Of  per  cent,  of  $225.40. 

18.  A  has  $852  deposited  in  the  bank,  and  wishes  to  draw 
out*,  5  per  cent,  of  it :  how  much  must  he  draw  for  ? 


10.  66|  per  cent,  of  420  cows. 

11.  105  per  cent,  of  850f  T. 

12.  116  per  cent,  of  875T9g- Ib. 

13.  241  per  cent,  of  $875.12|. 

14.  3.7J  per  cent,  of  $200. 
15.-  .33|  per  cent,  of  $687.24. 
16.  .87J  per  cent,  of 


257.  When  the  base  and  rate  are  given,  how  do  you  find  the 
percentage  ? 


PERCENTAGE.  221 

19.  A  merchant  has  1200  barrels  of  flour;   he  shipped  64>. 
per  cent,  of  it,  and  sold  the  remainder  :  how  much  did  he  sell  ? 

20.  A  merchant  bought  1200  hogsheads  of -molasses.     Oil 
getting  it  into  his  store,  he  found  it  short  3|  per  cent. :  how 
many  hogsheads  were  wanting  ? 

21.  What  is  the  difference  between  5|  per  cent,  of  $800, 
and  6J  per  cent,  of  $1050  ? 

22.  Two  men  had  each  $240.     One  of  them  spends  14  per 
cent.,  and  the  other  18J  per  cent.:   how  many  dollars  more 
did  one  spend  than  the  other  ? 

23.  A  man  ha.s  a  capital  of  $12500  ;  he  puts  15  per  cent, 
of  it  in  State  stocks,  33f  per  cent,  in  railroad  stocks,  and  25 
per  cent,  in  bonds  and  mortgages  :  what  per  cent,  has  he  left, 
and  what  is  its  value  ? 

24.  A  farmer  raises  850  bushels  of  wheat :   he  agrees  to 
sell  18  per  cent,  of  it,  at  $1.2s>"a  bushel ;   50  per  cent,  of  it, 
at  $1.50  a  bushel;   and  the  remainder,  at  $1.75  a  bushel: 
how  much  does  he  receive  in  all  ? 

258.    To  find  the  per  cent,  which  one  number  is   of  an- 
other. 

1.  What  per  cent  of  $16,  is  $4? 

ANALYSIS. — Since  the  percentage  is  equal 
to  the  base  multiplied  by  the  rate,  the  rate  OPERATION. 

will  be  equal  to  the  percentage  divided  by     4^16  =  .25,  or 
the  base.     In  this  example,  the  percentage  95  ~er  cenj. 

is  $4,  and  the  base  is  $16 ;  hence,  the  rate 
is  equal  to  $4  H-  $16  =  25  per  cent. :  Hence, 

Rule. — Divide  the  percentage  by  the  base,  and  the  quo- 
tient, in  decimals,  will  express  the  rate. 

Examples. 

1.  What  per  cent,  of  20  dollars  is  5  dollars  ? 

2.  Forty  dollars  is  what  per  cent,  of  eighty  dollars  ? 

3.  What  per  cent,  of  200  dollars  is  80  dollars  ? 

258.   How  do  you  find  what  per  cent,  one  number  is  of  another? 


222 


PERCENTAGE. 


4.  Ninety  bushels  of"  wheat  is  what  per  cent,  of  1800  bu.  ? 

5.  Nine  yards  of  cloth  is  what  per  cent,  of  870  yards  ? 
C.    Forty-eight  head  of  cattle  are  what  per  cent,  of  a  drove 

of  1600? 

7.  What  per  cent,  of  $75.25,  is  $8.621  ? 

8.  f  is  what  per  cent,  of  -f-  ? 

9.  What  per  cent,  of  16.1875  yd.  is  37f  yd.? 

10.  .875  is  what  per  cent,  of  .125? 

11.  15  is  what  per  cent,  of  65?  =tfa 

12.  27  is  what  per  cent,  of  35  ? 

13.  A  man  has  $550,  and  purchases  goods  to  the  amount 
01  $82.75 :  what  per  cent,  of  his  money  does  he  expend  ? 

14.  A  merchant  goes  to  New  York,  with  $1500 ;  he  first 
lays  out  20  per  cent.,  after  which   he   expends   $660  :    what 
per  cent,  was  his  last  purchase  of  the  money  that  remained 
after  his  first  ? 

15.  Out  of  a  cask  containing  300  gallons,  60  gallons  "are 
drawn :  what  per  cent,  is  this  ? 

16.  The  population  of  a  town,  in  a  certain  year,  was  5682  ; 
5  years  afterward,  it  was  7296 :  what  was  the  per  cent,  of 
increase  during  the  interval  ? 

17.  A  man  purchased  a  farm  of  75  acres,  at  $42.40  an 
acre ;   he  afterward  sold  the  same  farm  for  $3577.50 :   what 
was  his  gain  per  cent,  on  the  purchase-money  ? 

259.    Having   given  the  percentage   and  rate,  to  find  the 
base. 

1.   $750  is  18  per  cent,  of  what  number  of  dollars  ? 

ANALYSIS. — Since   the    percentage  is 

equal  to  the  product  of  the  base  by  the  OPERATION. 

rate,  the  base  is  equal  to  the  percent-       750-r-.18  =  $4166f. 
age  divided  by  the  rate:     Hence, 

Rule. — Divide  the  given  percentage  by  the  rate,  express?, 
decimally,  and  the  quotient  loill  be  the  base. 


259.  When  the  percentage  and  rate  are  given,  how  do  you  find 
the  base  ? 


COMMISSION.  223 

Examples. 

1.  960  is  25  per  cent,  of  what  number? 

2.  74  is  62|  per  cent,  of  what  number? 

3.  450  is  112  per  cent,  of  what  number? 

4.  Of  what  number  is  66,  §  per  cent.  ? 

5.  Of  what  number  is  1.75,  37J  per  cent.? 

6.  f  is  J  per  cent,  of  what  number  ? 

7.  In  a  school,  77  pupils   are  present,  which  is  87 J  per 
cent,  of  the  number  on  roll :  what  is  the  number  on  roll  ? 

8.  Suppose  the  population  of  a  town,  in  1855,  to  have 
been   15624,   which   was  56  per  cent,   of  its   population   in 
1860:   what  was  the  population  in  1860?    5.    , 

9.  In  a  mixture  of  wine  and  water,  there  are  16  gallons 
of  water,  which  is  80  per  cent,  of  the  whole  :    what  is  the 
amount  of  the  mixture  ?3_  & 

10.  A  and  B  are  in  partnership  ;  A  receives  of  the  profits, 
$370,  which  was  18|  per  cent,  of  the  whole  profit :  what  was 
the  total  profit,  and  what  was  B's  share  ? 


COMMISSION. 

260.  COMMISSION  is  an  allowance  made  to  an  agent  for  the 
transaction  of  business.     It  is  reckoned  at  a  certain  rate  per 
cent,  on  the  amount  of  money  employed. 

261.  To    find   the    percentage    of  Commission,   when   the 
rate  and  base  are  known. 

1.  What  is  the  commission  on  $4396,  at  6  per  cent.  ? 

ANALYSIS. — The  base  and  rate  being  OPERATION. 

given,  we  find  the  percentage  by  mul-        $4396 
tiplying  $4396  by  .06  (Art.  257).  .06 

Hence  the  $263.76  =  Commission 

Rule. 

Multiply  the  amount  employed  by  the  rate,  in  decimals, 
and  the  product  is  the  commission. 


224 


PERCENTAGE. 


Examples. 

1.  A  land  agent   sells   a  farm  for  $27560  :    what  is   the 
amount  of  his  commission,  at  5  per  cent.  ? 

2.  A  commission  merchant,  in  New  York,  received  from  St. 
Louis  a  quantity  of  flour,  which  he  sold  for  $5695  :  what  is 
the  amount  of  his  commission,  at  9|  per  cent.  ? 

3.  A  house  agent  collects  rents  to  the  amount  of  $1756.75; 
what  is  his  commission,  at  3  per  cent.,  and  what  amount  does 
he  pay  over  to  the  landlord  ? 

4.  A  factor  sells  60  bales  of  cotton  at  $425  per  bale,  and 
is  to  receive  2|  per  cent,  commission  :  how  much  must  he  pay 
over  to  his  principal  ? 

5.  A  commission  merchant  sells  goods  to  the  amount  of 
$8750,  on  which  he  is  to  be  allowed  2  per  cent. ;  but  in  con- 
sideration of  paying  the  money  over  before  it  is  due,  he  is  to 
receive  1 1  per  cent,  additional :  how  much  must  he  pay  over 
to  his  principal  ?  • 

6.  A  drover  agreed  to  take  a  drove  of  cattle  to  New  York, 
and  sell  them  on  a  commission  of  5  per  cent,  on  the  estimated 
value,  84250,  or  on  any  higher  sum  for  which  he  might  sell : 
he  sold  them  at  an  advance  on  their  estimated  value  of  10 
per  cent. :  what  commission  did  he  receive  ? 

7.  What  does  a  commission  merchant,  who  charges  5  per 
cent,  commission  and  2|  per  cent,  guarantee,  receive  on  a  bill 
of  goods  for  $2765.50? 

NOTE. — Guarantee  is  indemnity  against  risk. 

8.  A  real-estate  agent  purchased  a  house  for  $3650,  charging 
2J  per  cent,  commission.     In  the  course  of  a  few  days,  the 
value  of  the  property  advanced  15  per  cent.,  and  he  was  then 
directed  to  sell  the  property.    What  was  the  amount  of  his 
commission,  for  purchase   and  sale,  the  rate  being  the  same 
in  both  cases  ? 


260.  What  is  Commission? 

261.  How  do  you  find  the  commission,  when  the  base  and  rate 
are  known  ? 


COMMISSION.  225 

9.  I  directed  my  agent  to  purchase  <£5  lots  of  ground,  at 
$650  per  lot,  and  to  pay  the  expense   of  examining  titles, 
which  was  $5  per  lot :  what  did  the  lots  cost  me,  if  the  com' 
mission  was  4  per  cent.  ? 

10.  A  commission  merchant  was  allowed  5  barrels  of  flour 
commission  for  every  60  barrels  that  he  sold  :  what  was  his 
rate   of  commission,   and  how  much  did  he   receive   on  570 
barrels  ? 

11.  An  agent  was  employed  to  sell  a  ship,  whose  price  was 
$30000  :   if  he  sold  it  at  that  or  any  higher  price,  he  was  to 
receive  $850  :   he  sold  it  at  the  price  named  :  what  was  the 
rate  per  cent,  of  commission  ? 

262.  To  find  the  commission  and  base,  when  the  rate  and 
the  sum  of  the  base  and  commission  are  known. 

1.  Merchant  A  sent  to  B,  a  commission  merchant,  $3825, 
to  be  invested  in  the  purchase  of  flour ;  B  is  to  receive  2  per 
cent,  on  the  amount  paid  for  the  flour :  what  was  the  value 
of  the  flour,  and  what  was  the  commission  ? 

ANALYSIS. — Since  the  broker 

receives  2  per  cent.,  it  will  re-  OPERATION. 

quire  $1.02  to  purchase  1  dol-         1.02  )  3825.00  (  $3750  Ans. 
lar's  worth  of  flour ;  hence,  there  306 

will  be  as  many  dollars1  worth  ^    _ 

purchased  as>$1.02  is  contained  ^ 

times  in  $3825;  that  is,   $3750 
worth.    The  commission  will  be  510 

2  per  cent,  of  $3750,  or  $3825  510 

—  3750  =  $75:    Hence, 

Rule. 

I.  Divide  the  given  amount  by  1  plus  the  rate  of  com- 
mission, expressed  decimally,  and  the  quotient  will  be  the 
base  of  percentage. 

II.  Subtract  the  base  from  the  given  amount,  and  the  re- 
mainder will  be  the  commission. 


262.   How  do  you  find  the  commission  and  base,  when  the  rate 
and  sum  of  the  commission  and  base  are  known  ? 

10* 


226 


PERCENTAGE. 


Examples. 

1.  A  grocer  received  $750,  to  expend  in  the  purchase  of  flour. 
Allowing  7^  per  cent,  commission,  what  did  he  pay  for  the  flour? 

2.  A  merchant  at  Chicago  sends  to  his  agent  in  New  York 
$5413.05,  with  directions  to  buy  coffee,  and  to  charge  a  com- 
mission on  the  money  expended  of  3|  per  cent.  :    what  was 
the  amount  of  commission  ? 

3.  What  value  of  stock  can  be  purchased  for  $20119,  if  a 
commission  of  5|  per  cent,  be  allowed  on  its  cost  ? 

4.  How  many  barrels  of  flour,  at  $7  a  barrel,  can  be  bought 
for  $2657.20,  if  4  per  cent,  commission  be  allowed  on  the 
money  paid  ? 

5.  A  merchant  in  New  York  receives  from  Boston  $25000, 
to  be  expended  in  flour  ;  he  was  allowed  a  commission  of  2J 
per  cent,  on  the  money  paid  :  what  was  the  amount  of  com- 
mission ? 

6.  A  merchant  in  New  York  sends  $12600  to  a  commission 
merchant  in  Chicago  for  the  purchase  of  flour ;   the  latter 
charges  5  per  cent,  commission  :   what  amount  was  expended 
in  buying  the  flour,  and  what  was  the  commission  ? 

7.  A  commission  merchant  in  New  York  received  $5000 
from  Cincinnati  for  purchasing  dry  goods  ;  he  charged  a  com- 
mission of  3  per  cent,  on  what  he  paid  :  what  commission  .was 
received  ? 

8.  In  a  given  time,  a  sum   of  money,   placed  at  interest, 
had  increased  17  per  cent.,  and  amounted  to  $5679.45  :  what 
was  the  sum  of  money  at  interest  ? 

9.  A  merchant  is  directed  to  expend  $5642.48  in  the  pur- 
chase of  cloth  ;  he  is  allowed  a  commission  of  2|  per  cent,  on 
what  he  pays  for  the  cloth,  and  charges  in  addition  1|  per 
cent,  for  storage  :  how  much  did  he  lay  out  in  cloth  ? 

10.  A  merchant  in  New  Orleans  received  from  New  York 
$21630,  with  orders  to  invest  in  cotton,  allowing  a  commission 
of  2|.per  cent.  ;  marine  insurance,  1J  per  cent.;  cartage  and 
freight,   1J  per  cent.  :   what  amount  was  laid  out  in  buying 
cotton,  and  how  much  was  bought,  at  15  cents  per  Ib.  ? 


PROFIT   AND    LOSS.  227 


PROFIT   AND    LOSS. 

263.  PROFIT  AND  Loss   arc   commercial  terms,   indicating 
gain  or  loss   in  business  transactions.     The  gain  or  loss  is 
always  estimated  on  the  cost  price. 

264.  To  find  the  gain  or  loss,  when  the  cost  and  selling 
price  are  given. 

1.  Bought  a  piece  of  cloth,  containing  75  yards,  at  $5.25 
per  yard,  and  sold  it  at  $5.75  per  yard :  how  much  was 
gained  in  the  trade  ? 

OPERATION. 

ANALYSIS.— Subtract  the  en-  $5^5  x  75  -  $39375 

tire  cost  from  the  entire  selling  5  ^5  x  ^5  _  g  t;.  |  0  -, 

price,  and  the  remainder  will  . '            Q«  ?,  ._  '" 
be  the  gain;   or  multiply  the 
gain  on  1  yard  by  the  number 

Of  yards  :    Hence,  5.75-5.25  =  .50  gam  on  1  yd. 

.50x75  =  $37.50  =  whole  gain. 

Rule. — Find  the  difference  between  the  cost  and  selling 
price,  and  the  remainder  will  be  the  gain  or  loss. 

Examples. 

1.  A  merchant  bought  a  horse  for  $175.50,  and  sold  it  for 
$21 5| :  what  was  the  gain? 

2.  A  merchant  bought  a  ship  for  $7500,  and  paid  $1900 
for  repairs  ;  he  then  sold  it  for  $12000  :  what  was  the  gain? 

3.  Bought  a  hogshead  of  brandy  at  $1.25  per  gallon,  and 
sold  it  for  $78  :   was  there  a  loss  or  gain? 

265.  To  determine  the  selling  price,  when  the  cost  and 
gain  or  loss  are  known. 

1.  Bought  a  piece  of  calico,  containing  56  yards,  at  27 
cents  a  yard  :  what  must  it  be  sold  for  a  yard,  to  gain.  $2. 24? 

263.  What  do  the  terms  Profit  and  Loss  indicate  ?  On  what  is 
the  gain  or  loss  estimated  ? 


228  PERCENTAGE. 

OPEKATION. 

ANALY8is.-First    find    the  56  7ards>  at  27  ce»ts  =  ^.12 

cost,  then  add  the  profit,  and  Profit      ....       2.24 

divide  the  sum  by  the  num-  jt  must  geu  for     >        _  $17.36 
ber  of  yards.  56)17.36 

31  cts.  a  yd. 

Rule. — Add  the  gain  to  the  cost,  or  deduct  the  loss,  and 
the  sum  or  difference  will  be  the  selling  price. 

Examples. 

1.  If  a  hogshead  of  wine  cost  $159  :  for  what  must  it  be 
sold,  per  pint,   that  there   may  be   a  gain  of  50  cents  per 
gallon  ? 

2.  A  house  cost  $475&:  for  what  must  it  be  sold,  that 
the  owner  may  realize  $604),  after  paying  his  agent  a  commis- 
sion of  $65  ? 

3.  A  merchant,  in  selling  500  bushels  of  corn,  which  cost 
$236,  lost  $45:  what  did  he  obtain  per  bushel? 

4.  For  what  must  a  farm,  which  cost  $7960,  be  sold,  that 
the   owner  may  realize   $1800,  after  paying  to  his  agent  a 
commission  of  $50  ? 

266.   To  find  the  gain  or  loss,  when  the  cost  and  rate  per 
cent,  of  gain  or  loss  are  known.  (Rule,  Art.  257.) 

Examples. 

1.  What  is  the  gain  on  $3750,  at  6  per  c§pt.  ? 

2.  What  would   be  the  gain  hi  the   sale  of  a  "house  for 
$12750,  at  18|  per  cent.  ? 

3.  A  gentleman  lost,  in  the  sale  of  bank-stock  that  cost 
$24760,  6  per  cent. :   what  was  the  loss  ? 


264.  How  do  you  find  the  gain  or  loss,  when  the  cost  and  selling 
price  are  given  ? 

265.  How  do  you  find  the  selling  price,  when  you  know  the  cost 
and  loss  or  gain  ? 

266.  How  do  you  find  the  gain  or  loss,  when  the  cost  and  rato 
are  known  ? 


PROFIT   AND   LOSS  229 

4.  Bought  a  piece  of  cassimcre,  containing  28  yards,  at  1J 
dollar  a  yard  ;  but  finding  it  damaged,  am  willing  to  sell  it  at 
a  loss  of  15  per  cent.  :  how  much  must  be  asked  per  yard? 

5.  A  merchant  purchased  3275  bushels  of  wheat,  for  which 
he  paid  13517.10;  but  finding  it  damaged,  is  willing  to  lose 
10  per  cent.  :  what  must  it  sell  for  per  bushel? 

6.  Bought  50  gallons  of  molasses,  at  75  cents  a  gallon, 
10  gallons  of  which  leaked  out.     At  what  price,  per  gallon, 
jiust  the  remainder  be  sold,  that  I  may  clear  10  per  cent,  on 
the  cost  of  the  whole  ? 

7.  A  merchant    buys   158   yards   of  calico,  for  which  he 
pays  20  cents  per  yard  ;   one-half  is   so  damaged  that  he  is 
obliged  to  sell  it  at  a  loss  of  6  per  cent.  ;  the  remainder  he 
sells  at  an  advance  of  19  per  cent.  :  how  much  did  he  gain? 

267.    The  cost  and   selling  price   being  known,  to   deter- 
mine the  rate  per  cent,  of  gain  or  loss. 

1.    If  I  buy  coffee  at  16  cents,  and  sell  it  at  20  cents  a 
pound,  how  much  do  I  make  per  cent.  ? 

ANALYSIS.—  The    gain    is    4    cents.     The  OPERATION. 

gain  divided   by  the'  cost,  gives   the   rate 
(Art.  258).  4  -H  16  =  .25. 

Examples. 

1.  A  man  bought  a  house  and  lot  for  $1850.50,  and  sold 
them  for  $1517.41  :  how  much  per  cent,  did  he  lose? 

2.  A  'merchant  bought  650  pounds  of  cheese  at  10  cents 
per  pound,  and  sold  it  at   12  cents  per  pound:   how  much 
did  he   gain  on  the  whole,  and  how  much  per  cent,  on  the 
money  laid  out  ? 

3.  If  I  sell  a  piano,  which  cost  $275,  for  $315,  what  was 
the  rate  per  cent,  of  gain  ? 

4.  A  herd  of  cattle  was  bought  in  Kentucky,  ut  an  ex- 
pense  of  $3750  ;   the   cost  of  transportation  was-  $250  ;   it 
was  sold  in  New  York,  for  $5725:  what  was  the  rate  per 
cent,  of  gain,  after  paying  the  expense?   L  \)ty  ^  ^f 


267.  How  do  you  find  the  rate,  when  the  cost  and  selling  price 
are  known  ? 


230  PERCENTAGE. 

268.    To   find   the    cost,  when   the   selling   price   and   rate 
per  cent,  of  gain  or  loss,  are  known. 

1.  I  sold  a  parcel  of  goods  for  $195.50,  on  which  I  made 
15  per  cent.:   what  did  they  cost  me"? 

ANALYSIS.  —  1  dollar  of  the   cost  plus   15          OPERATION. 
per  cent.,  will  be  what  that  which  cost  $1        1.15)1  95.  50 
sold  for,  viz.,  $1.15  :  hence,  there  will  be  as  T~7     ~ 

many  dollars  of  cost,  as  $1.15  is  contained 
times  in  what  the  goods  brought. 

2.  If  I  sell  a  parcel  of  goods  for  $170,  by  which  I  lose 
15  per  cent.,  what  did  they  cost  ? 

ANALYSIS.  —  1  dollar  of  the   cost  less   15         OPERATION 

* 


per  cent,  will  be  what  that  which  cost  1 
dollar   sold    for,    viz.,    $0.85:    hence,   there 


N  i  70 


were  as  many  dollars  of  cost,  as  .85  is  con-  $200    Ans. 

tained  times  in  what  the  goods  brought. 
Hence,  to  find  the  cost, 

Rule.  —  Divide  the  amount  received  by  1  plus  the  per 
cent,  when  there  is  a  gain,  and  by  1  minus  the  per  cent. 
when  there  is  a  loss,  and  the  quotient  will  be  the  cost. 

* 

Examples.    * 

1.  A  carriage  was  sold  for  $350,  by  which  a  gain  of  25 
per  cent,  was  made  :  what  was  the  cost  ?    V  V^ 

2.  If  I  sell  a  parcel  of  goods  for  $170,  by  which  I  lose 
15  per  cent.,  what  did  they  cost  ? 

3.  A  cargo   of  wheat  was   sold  for  $12500,  by  which  a 
gain  of  25  per  cent,  was   made  :   what  was  the   amount   of 
net  gain,  after  paying  $150  for  freight,  and  $75  for  other 
charges?     \*v  T^ 

4.  A  commission  merchant  sold.  a  lot  of  iron,  which  had 
been  consigned  to  him,  for  $25600,  by  which  a  gain  of  31 
per  cent,  on  the  invoice  was  made  :  allowing  him  5  per  cent. 
commission,  what  was  the  net  gain  ? 


268.  How  do  you  find  the  cost,  when  the  selling  price  and  rate 
are  known  ? 


INSURANCE.  231 

( 

INSURANCE. 

269.  INSURANCE  is  an  obligation,  generally  in  writing,  by 
which  individuals  or  companies  bind  themselves  to  indemnify 
the  owners  of  certain  property,  such  as  ships,  goods,  houses, 
&c.,  from  loss  or  hazard. 

270.  The  BASE  of  insurance,  is  the  amount  for  which  the 
property  is  insured. 

271.  "The   POLICY  is  the  written  agreement   made  by  the 
,     parties. 

272.  PREMIUM  is  the  amount  paid  by  him  who  owns  the 
property,  to  those  who  insure  it. 

273.  To   find   the   premium,  -when  the  base  and  rate  are 
known. 

Rule. — Same  as  in  Art.  257. 

Examples. 

1.  What   is   the   premium   for   the   insurance,   of  a  house 
valued  at  $8754,  against  loss  by  fire,  for  one  year,  at  \  per 
cent.  ? 

2.  What  would  be  the  premium  for  insuring   a   ship  and 
cargo,  valued  at  $37500,  from  New  York   to   Liverpool,  at 
3|  per  cent.? 

3.  What   would   be   the   insurance    on   a   ship   valued   at 
$47520,  at  J  per  cent.?     Also,  at  \  per  cent.? 

4.  A  merchant  wishes  to  insure  on  a  vessel  and  cargo  at 
sea,   valued  at   $28800  :    what  will  be   the  premium,  at   1| 
per  cent.? 

269.  What  is  Insurance  ?— 270.  What  is  the  base  of  insurance  ? 
271.  What  is  the  policy?— 272.  What  is  the  premium? 
273.  How  do  you  find  the  premium,  when  the  base  and  rate  are 
known  ? 


232  PERCENTAGE. 

5.  A   merchant   owns   three-fourths    of  a   ship  valued   at 
$24000,  and  insures  his  interest  at  2£  per  cent.:   what  doea 
he  pay  for  his  policy  ? 

6.  A  merchant  learns  that  his  vessel  and  cargo,  valued  at 
$36000,    have   been  injured   to   the   amount  of  $12000  ;    he 
effects  an  insurance  on  the  remainder,  at  5|  per  cent. :  what 
premium  does  he  pay? 

7.  My  furniture,  worth  $3440,  is  insured  at  2f  per  cent.; 
my  house,   worth  $10000,   at   1J   per   cent.;    and  my  barn, 
horses,  and  carriages,  worth  $1500,  at  3  J  per  cent. :  what  is 
the  whole  amount  of  my  insurance  ? 

8.  A  merchant  imported  250   pieces  of  broadcloth,  each 
piece  containing  36|  yards,  at  $3.25  a  yard ;  he  paid  41  per 
cent,  insurance   on  the   selling   price,   $4.50   a  yard :    if  the 
goods  were   destroyed   by   fire,   and  he   got   the   amount   of 
insurance,  how  much  did  he  make  ? 

9.  A  vessel  and'  cargo,  worth  $65000,  are  damaged  to  the 
amount  of  20  per  cent.,  and  there  is  an  insurance  of  50  per 
cent,  on  the  loss :   how  much  will  the  owner  receive  ? 


STOCKS  AND  BROKERAGE. 

274.  A  CORPORATION  is  a  collection  of  persons,  authorized 
by  law  to  do  business  together. 

275.  A    CHARTER  is   the   law  which   defines   their  rights, 
powers,  and  duties. 

276.  CAPITAL,  or  STOCK,  is  the  money  paid  in  to  carry  on 
the  business  of  the  corporation. 

277.  STOCKHOLDERS  are  the  individuals  composing  the  cor- 
poration. 

278.  SHARES    are    portions    of    the    stock    owned    by   the 
stockholders. 

279.  CERTIFICATES  are  the  written  evidences  of  the  owner- 
ship of  stock. 


STOCKS   AND   BROKERAGE.  233 

280.  UNITED  STATES  STOCKS,  or  STATE  STOCKS,  are  the  bonds 
of  the  United  States,  or  of  a  State,  bearing  a  fixed  interest. 

281.  The  PAR  VALUE  of  a  stock,  is  the  number  of  dollars 
named   in   each   share.     Shares   are   usually  of  8100   each ; 
sometimes  $50,  and  sometimes  $25. 

282.  The  MARKET  VALUE  of  a  stock,  is  what  the   stock 
brings  per  share,  when  sold  for  cash. 

283.  PREMIUM  is  the  rate  per  cent,  which  a  stock   sells 
for,  above  its  par  value. 

284.  DISCOUNT  is  the  rate  per  cent,  which  a  stock  sells 
for,  below  its  par  value. 

285.  BROKERAGE  is  an  allowance  made  to  an  agent  who 
buys  or  sells  stock,  uncurrent   money,  or  bills   of  exchange. 
The  brokerage,  in  the   city  of  New  York,  is  generally  one- 
fourth  per  cent,  on  the  par  value  of  the  stock. 

286.  To  find  the  market  value  of  stock,  when  at  a  pre- 
mium or  discount. 

1.  What  is  the  value  of  150  shares  of  Erie  stock,  par  100, 
which  is  selling  at  16  per  cent,  discount  ? 

ANALYSIS.— 150  shares  are  nominally  worth  $15000;  100—16=84, 
whic.li  is  the  rate  per  cent,  to  be  taken  of  the  base,  $15000. 

Rule. — Multiply  the  nominal  value  by  the  rate  per  cent., 
to  be  taken  of  the  base,  and  the  product  will  be  the  market 
value. 

2.  What  is  the  market  value  of  200  shares  of  bank  stock, 
par  at  50,  which  is  selling  at  20  per  cent,  premium  ? 

274.  What  is  a  corporation? — 275.  What  is  a  charter? 

276.  What  is  capital,  or  stock  ?— 277.  What  are  stockholders  ? 

278.  What  are  shares?— 279.  What  are  certificates? 

280.  What  are  United  States  or  State  stocks? 

281.  What  is  par  value?— 282.  What  is  market  value? 
283.  What  is  premium? — 284.  What  is  discount? 

285.  What  is  brokerage? 

286.  How  do  you  find  the  market  value  of  stock  ? 


234  PERCENTAGE. 

3.  How  much  must  be  paid  for  $25600  of  stock,  which  is 
selling  at  87f  per  cent.  ? 

4.  A  broken  bank  has  a  circulation  of  $98000,  and  pur- 
chases the  bills  at  85  per  cent. :   how  much  is  made  by  the 
operation  ? 

5.  A  broker  sells   $50000   of  stock  on  commission,  at 
per  cent. :   what  is  the  brokerage  ? 

6.  What  must  be  paid  for  175  shares   of  Hudson  River 
Railroad  stock,  par  100,  which  is  selling  at  9  per  cent,  dis- 
count, if  the  brokerage  be  J  per  cent.  ? 

7.  A  gentleman  directs  a  broker  to  purchase  250  shares 
of  bank  stock,  par  100,  which  is  selling  at  8  per  cent,  pre- 
mium :  what  is  its  cost,  if  the  brokerage  be  at  f-  per  cent.  ? 

287.    To  find  how  much  stock,  at  par  value,  can  be  pur- 
chased for  a  given  sum. 

1.  What  amount  of  stock,  at  par  value,  can  be  purchased 
for  $12192,  when  it  is  at  5  per  cent,  discount,  if  1  per  cent, 
be  charged  for  brokerage  ? 

ANALYSIS. — Since  the   stock  is   at  5  OPERATION. 

per  cent,  discount,  $1  of  it  would  cost  o,5_j_  QI__  95")  12192 

95  cents :  adding  the  brokerage,  it  will  - — r — 
cost  96  cents:    Hence, 

Rule. — Divide  the  given  sum  by  the  cost  of  I  dollar  of 
the  stock,  plus  the  brokerage. 

2.  What  amount  of  government  stock  can  I  buy  for  $15525, 
when  it  is  selling  at  3|  per  cent,  premium  ? 

3.  How  many  shares  of  bank  stock,  par  25,  can  be  bought 
for  $2730,  when  it  is  selling  at  5  per  cent,  premium  ? 

4.  A  broker  is  authorized  to  expend  $20450  in  purchasing 
N.  Y.  State  stocks,  which  are  selling  at  2  per  cent,  premium : 
what  amount  of  stock  does  he  buy,  after  allowing  £  per  cent, 
brokerage  ? 

5.  Erie  Railroad  stock  is  selling  at  24  per  cent,  discount, 
and  brokerage  is  charged  at  f  per  cent. :   how  many  shares 
can  be  bought  for  $9195? 

287.  How  do  you  find  the  amount  of  stock  which  can  be  pur 
chased  for  a  given  sum? 


INTEREST.  235 


INTEREST. 

288.  INTEREST  is  a  payment  for  the  use  of  money. 

289.  PRINCIPAL  is  the  money  on  which  interest  is  paid. 

290.  The  RATE  of  interest,  is  the  per  cent,  paid  for  1  year. 

291.  AMOUNT  is  the  sum  of  the  principal  and  the  interest. 
Interest  is  always  reckoned  at  a  certain  rate,  by  the  year, 

or  per  annum. 

In  interest,  by  general  custom,  a  year  consists  of  12 
months,  each  having  30  days  ;  hence,  in  a  year,  for  com- 
puting interest,  there  are  360  days. 

In  almost  every  country  and  State,  the  rate  of  interest  is 
fixed  by  law,'  and  is  called,  the  Legal  Eate.  This  rate  differs 
in  different  States  and  countries. 

Any  rate  above  the  legal  rate,  is  usury,  which  is  forbidden 
by  law. 

CASE    I. 

292.  To   find   the    interest   on    any  principal   for   one   or 
more  years. 

1.  What  is  the  interest  of  $1960,  for  4  years,  at  7  per 
cent.  ? 

ANALYSIS. — The   principal   is    the  OPERATION. 

base,   and  the   interest  is   the   per-  $1960 

centage,  which  is  found  by  multiply-  .07,  rate 

ing  the  principal  by  the  rate;  there-  , «*  on   inf   fnr.  i  „ 

fore,    $137.20   is  the   interest  for  1  137'20'  ^  f°r  l  ^' 

year,  and  this  interest  multiplied  by          i»  No'  of  vears- 

4,   gives   the   interest   for  4  years:  $548.80 
Hence, 

Rule. — Multiply  the  principal  by  the  rate,  expressed 
decimally,  and  the  product  by  the  number  of  years. 


288.  What  is  Interest  ?— 289.  What  is  principal  ?— 290.  What  is 
rate  of  interest ?— 291.  What  is  amount?  What  does  per  annum 
mean  ?  What  is  legal  interest  ? 

292.  How  do  you  find  the  interest  of  any  principal,  for  any  num- 
ber of  years  ?  Give  the  analysis. 


236  PERCENTAGE. 

Examples. 

1.  What  is  the  interest  of  $650,  for  one  year,  at  6  per  cent.? 

2.  What  is  the  interest  of  $950,  for  4  years,  at  7  per  cent.  ? 

3.  What  is  the  amount  of  $3675  in  3  years,  at  7  per  cent.  ? 

4.  What  is  the  amount  of  $459  in  5  years,  at  8  per  cent.  ? 

5.  What  is  the  interest  of  $211.26,  for  1  year,  at  4J  per  ct.  ? 

6.  What  is  the  interest  of  $1576.91,  for  3  yr.,  at  7  per  ct.? 

7.  What  is  the  amount  of  $957.08  in  6  years,  at  3|  per  ct.  ? 
r — 8.  What  is  the  interest  of  $375.45,  for  7  years,  at  J  per  ct.? 
*     9.  What  is  the  amount  of  $4049.87  in  2  years,  at  5  per  ct.  ? 

10.  What  is  the  amount  of  $16199.48  in  16  yr.,  at  5|  per  ct.  ? 

NOTE. — When  there  are  years  and  months,  and  the  months  are 
aliquot  parts  of  a  year,  multiply  the  interest  for  1  year  l>y  the 
years  and  months  reduced  to  the  fraction  of  a  year. 

11.  What  is  the  interest  of  $326.50,  for  4  years  and  2 
months,  at  7  per  cent.  ? 

12.  What  is  the  interest  of  $437.21,  for  9  years  and  3 
months,  at  3  per  cent.  ? 

13.  What  is  the  amount  of  $1119.48,  after  2  years  and 
6  months,  at  7  per  cent.  ? 

14.  What  is  the  amount  of  $179.25,  after  3  years  and  4 
months,  at  7  per  cent.  ? 

15.  What  is  the  amount  of  $1046.24,  after  3  months,  at 
5|  per  cent.? 

16.  What  is  the  amount  of  $6704.25,  after  1  year  and  4 
months,  at  6|  per  cent.? 

17.  What  is  the  interest  of  $3750.87,  for  2  years  and  9 
months,  at  8  per  cent.  ? 

CASE    II. 

293.  To  find  the  interest  on  a  given  principal  for  any  rate 
and  time. 

1.  What  is  the  interest  of  $876.48,  at  6  per  cent.,  fo. 
4  years  9  months  and  14  days  ? 


293.   How  do  you  find  the  interest  for  any  time,  at  any  rate? 


INTEREST.  237 

ANALYSIS. — The  interest  for  1  year  is  the  product  of  the  prin- 
cipal and  the  rate.  If  the  interest  for  1  year  be  divided  by  12, 
the  quotient  will  be  the  interest  for  1  month ;  if  the  interest  for 
1  month  be  divided  by  30,  the  quotient  will  be  the  interest  for 
1  day. 

The  interest  for  4  years  is  4  times  the  interest  for  1  year ;  the 
interest  for  9  months,  9  times  the  interest  for  1  month ;  and  the 
interest  for  14  days,  14  times  the  interest  for  1  day. 

OPERATION. 

$876.48 
.06. 


12)52.5888=int.  for  1  yr.    52.5888    x   4=$210.3552  4yr. 
30 )  4.3824 =int.  for  1  mo.     4.3824   x    9=$  39.4416  9  mo. 
.H608=int.  for  1  da.       .14608  x  14=$     2.0451   14  da. 

Total  interest,     $251.8419+  S 

Rule. 

I.  Find  the  interest  for  1  year : 

II.  Divide  this  interest  by  12,  and  the  quotient  will  be  the 
interest  for  1  month: 

III.  Divide  the  interest  for  1  month  by  30,  and  the  quo- 
tient will  be  the  interest  for  1  day: 

IV.  Multiply  the  interest  for  1  year  by  the  number  of 
years,  the  interest  for  1  month  by  the  number  of  months,  and 
the  interest  for  1  day  by  the  number  of  days,  and  the  sum 
of  the  products  will  be  the  required  interest. 

Second  Rule. 

294.   There  is  another  rule  resulting  from  the  last  analysis, 
which  is  a  good  general  rule  for  computing  interest. 

I.  J?ind  the  interest  for  1  year,  and  divide  it  by  12  :  the 
quotient  will  be  the  interest  for  1  month  : 

II.  Multiply  the  interest  for  1  month  by  the  time  expressed 
in  months  and  tenths  of  a  month,  and  the  product  will  be  the 
required  interest. 

NOTE. — Since  a  month  is  reckoned  at  30  days,  the  quotient  of 
any  number  of  days,  divided  by  B,  will  be  tenths  of  a  ir.onth. 


233 


PERCENTAGE. 


1.  What  is  the  interest  of  $327.50,  for  3  years  7  montl 
and  13  days,  at  7  per  cent.? 


OPERATION. 


3  years  —  36  mos. 
7  mos. 
13  days  =      .4^  mos. 

Time  =  43.4|  mos. 

NOTE. — The  method  em- 
ployed, and  the  number  of 
decimal  places  used,  in  com- 
puting interest,  may  affect  the 
mills,  and,  possibly,  the  last 
figure  in  cents.  It  is  best  to 
use  5  places  of  decimals. 


$327.50 
.07 

12)22.9250     =int.  for  1  year. 

1.9 104  +  =  int.  for  1  month. 
43. 4|  —  time  in  months. 

.6368 

76416 

57312 

76416 


$82.97504  Ans. 


Examples. 

1.  What  is  the  interest  of  $132.26,  for  1  year  4  months 
and  10  days,  at  6  per  cent,  per  annum  ? 

2.  What  is  the  interest  of  $25.50,  for  1  year  9  months  and 
12  days,  at  6  per  cent.? 

3.  What  is  the  interest  of  $1728.60,  at  7  per  cent.,  for 

2  years  6  months  and  21  days  ? 

4.  What  is  the  interest  of  $288.30,   at   7  per  cent.,  for 

I  year  8  months  and  27  days  ? 

5.  What  is  the  interest  of  $576.60,  at  6  per  cent.,  for 
10  months  and  18  days  ? 

6.  What  is  the  interest  of  $854.42,  at  6  per  cent.,  for 

3  months  and  9  days  ? 

7.  What  is  the  interest  of  $1123.20,  at  6  per  cent.,  for 

II  months  and  6  days  ? 

8.  What  is  the  interest  of  $2306.54,  at  5  per  cent.,  for 
7  months  and  28  days  ? 

9.  What  is  the  interest  of  $4272.10,  at  5  per  cent.,  for 
10  months  and  28  days  ? 

294.   How  do  you  find  the  interest  for  years,  months,  and  days, 
by  the  second  method? 


INTEREST.  239 

10.  What  is  the  interest  of  $1620,  at  4  per  cent.,  for  5 
years  and  24  days  ? 

11.  What  is  the  interest  of  $2430.72,  at  4  per  cent.,  for 
10  years  and  4  months  ? 

12.  What  is  the  interest  of  $3689.45,  at  7  per  cent.,  for 
4  years  and  7  months  ? 

13.  What  is  the  interest  of  $2945.96,  at  7  per  cent.,  for 
7  years  and  3  days  ? 

14.  What  is  the  interest,  at  8  per  cent.,  of  $675.89,  for 
3  years  6  months  and  6  days  ? 

15.  What  is  the  interest  of  $648.54,  for  7  years  6  months, 
at  4 1  peV  cent.  ? 

16.  What  is  the  interest  of  $1297.10,  for  8  years  5  months, 
at  5J  per  cent.  ? 

17.  What  is  the  interest   of  $864.768,  for  9  months   25 
days,  at  6J  per  cent.  ? 

18.  What  is  the  interest  of  $2594.20,  for  10  months  and 
9  days,  at  7J  per  cent.  ? 

19.  What  is  the  amount  of  $2376.84,  for  3  years  9  months 
and  12  days,  at  8J  per  cent.  ? 

20.  What  is  the  amount  of  $5148,40,  for  7  years  11  months 
and  23  days,  at  9J  per  cent.  ? 

21.  What  is  the  amount  of  $3565.20,  for  3  years  9  months, 
at  10  J  per  cent.  ? 

22.  A  person  bought  a  bill  of  goods  amounting  to  $750, 
but  delayed  its  payment  for  6  months  and  13  days  after  it 
was  due  :   allowing  7  per  cent,  interest,  what  amount  should 
he  pay  ? 

23.  A  merchant  bought  flour  to  the  amount  of  82560,  and 
kept  it  in  store  4  months  15  days  before  he  sold  it:   he  then 
sold  the  whole  for  $4250.     Allowing  interest  at  7  per  cent., 
what  was  the  gain  ? 

24.  What  amount  is  necessary  to  discharge  a  mortgage  of 
$5675,  the   interest   of  which,  at  6  per  cent.,  has   not  been 
paid  for  3  years  9  months  22  days  ? 

25.  A  merchant  sold   flour  which  cost  him  $4912,  at  an 
advance  of  30  per  cent. ;   but  ho   sold  the  flour  on  a  credit 


240  PERCENTAGE. 

of  3  months,  and  had  kept  it  in   store  2  months  15  days : 
allowing  7  per  cent,  interest,  what  did  he  gain  ? 

26.  What  is  the  amount  of  $256,  for  10  months  15  days, 
at  7 1  per  cent.? 

27.  What  is  the  interest  on  a  note  of  $264.42,  given  Jan- 
uary 1st,  1852,  and  due  Oct.  10th,  1855,  at  4  per  cent.? 

28.  Gave  a  note  of  $793.26,  April  6th,  1850,  on  interest 
at  7  per  cent.:  what  is  due  September  10th,  1852? 

29.  What  amount  is  due  on  a  note  of  hand  given  June 
7th,  1850,  for  $512.50,  at  6  per  cent.,  to  be  paid  Jan.  1st, 
1851? 

30.  What  is  the  interest  on  $1250.75,  for  90  days,  at  10 
per  cent.  ? 

31.  What  is  the  amount  of  $7109,  from  Feb.  8th,  1848, 
to  Dec.  7th,  1852,  at  6|  per  cent.? 

32.  What  will  be  due  on  a  note  of  $213.27,  on  interest 
after   90   days,  at   7  per  cent.,  given  May  19th,  1836,  and 
payable  October  16th,  1838? 

33.  What  is  the  interest  of  $2132.70,   from   Nov.  17th, 
1838,  to  Feb.  2d,  1839,  at  7|-  per  cent.? 

34.  What  is  the  interest  of '$38463,  from  April  27th,  1815, 
to  Sept.  2d,  1824,  at  8  per  cent.? 

35.  What  is   the  interest   of  $14231.50,  from  June  29th, 
1840,  to  April  30th,  1845,  at  8|  per  cent.? 

36.  What  is  the  interest  of  $426.40,  from  Sept,  4th,  1843, 
to  May  4th,  1849,  at  9  per  cent.? 

37.  What  is  the  interest  of  $4320,  from  Dec.  1st,  1817, 
to  Jan.  22d,  1818,  at  9|  per  cent.? 

38.  What  is   the   amount  of  $397.16,   from  March  24th, 
1824,  to  March  31st,  1835,  at  10|  per  cent.? 

39.  What  is  the  amount  of  $164.60,  from  Sept.  27th,  1845, 
to  March  24th,  1855,  at  1J  per  cent.? 

CASE     III. 
295.  When  the  principal  is  in  pounds,  shillings,  and  pence. 

1.  What  is  the  interest,  at  7  per  cent.,  of  £27  15s.  9d., 
for  2  years  ? 


PARTIAL   PAYMENTS.  241 

OPERATION. 

ANALYSIS — The  interest  on  pounds  £27  15s.  9d.  —  27.7875 

and  decimals  of  a  pound  is  found  in  .07 

the    same    way   as    the   interest   on  1  945125 
dollars    and    decimals    of   a   dollar: 
after  which  the  decimal  part  of  the 

interest  may  be  reduced  to  shillings  £3.890250 

and  pence:    Hence,  £.89025  =  17s.  9fd. 

Rule  AnS'  £3  17s>  9*d' 

I.  Reduce  the  shillings  and  pence  to  the  decimal  of  a 
pound,  and  annex  the  result  to  the  pounds : 

II.  Find  the  interest  as  though  the  sum   were  United 
States  Money,  after  which  reduce  the  decimal  part  to  shil- 
lings and  pence. 

Examples. 

1.  What  is  the  interest  of  £67  19s.  6d.,  at  6  per  cent., 
for  3  years  8  months  16  days  ? 

2.  What  is  the  interest  of  £127  15s.  4d.,  at  6  per  cent., 
for  3  years  and  3  months  ?     • 

3.  What  is  the  interest  of  £107  16s.  10d.,  at  7  per  cent., 
for  3  years,  6  months,  and  6  days  ? 

4.  What  will  £279  13s.  8d.  amount  to,  in  3  years  and  a 
jalf,  at  5J  per  cent,  per  annum? 


PARTIAL  PAYMENTS. 

296.  A  PARTIAL  PAYMENT  is  a  payment  of  a  part  of  the 
amount  due  on  a  note  or  bond. 

We  shall  give  the  rule  established  hi  New  York  (see  John- 
son's Chancery  Reports,  vol.  i.,  page  17),  for  computing  the 
interest  on  a  bond  or  note,  when  partial  payments  have  been 


295.   How  do  you  find  the  interest,  when  the  principal  is  in 

pounds,  shillings,  and  pence? 

11 


242  PARTIAL    PAYMENTS. 

made.     The  same  rule  is  also -adopted  in  Massachusetts,  and 
in  most  of  the  other  States. 

Rule. 

I.  Compute  the  interest  on  the  principal  to  the  time  of 
the  first  payment,  and  if  the  payment  exceeds  fhis  interest, 
add  the  interest  to  the  principal,  and' from  the  sum  subtract 
the  payment :  the  remainder  forms  a  new  'principal : 

II.  But  if  the  payment  is  less  than  the  interest)  take  no 
notice  of  it  until  other  payments  are  made,  which  in  all 
shall  exceed  the  interest   computed  to  the  time  of  the  last 
payment:  then  add  the  .interest,  so  computed,  to  the  prin- 
cipal, and  from  the  sum  subtract  the  sum  of  the  payments: 
the  remainder  will  form  a  new  principal,  on  which  interest 
is  to  be  computed  as  before. 

NOTE. — In  computing  interest  on  notes,  observe  that  the  day 
on  which  a  note  is  dated  and  the  day  on  which  it  falls  due,  are 
not  both  reckoned  in  determining  the  time,  ~but  one  of  them  is 
always  excluded.  Thus,  a  note  dated  on  the  first  day  of  May, 
and  falling  due  on  the  16th  of  June,  will  bear  interest  but  one 
month  and  15  days. 

Examples. 

BUFFALO,  May  1st,  1826. 

1.  For  value  received,  I  promise  to  pay  James  Wilson  or 
order,  three  hundred  and  forty-nine  dollars,  ninety-nine  cents, 
and  eight  mills,  with  interest  at  6  per  cent. 

JAMES  PAYWELL. 

On  this  note  were  indorsed  the  following  payments : 
Dec.   25th,  1826,     received     $49.998 
July   10th,  1827,       .       .  4.998 

Sept.    1st,    1828,       .       .         15.008 
June  14th,  1829,       .       .          99.999 

What  was  due,  April  15th,  1830? 

296.  What  is  a  Partial  Payment?  What  is  the  rule  for  com- 
puting interest,  when  there  are  partial  payments? 


0 


PARTIAL   PAYMENTS.  243 

Principal  on  interest,  from  May  1st,  1826,      .        .       $349.998 
Interest  to  Deo.  25th,  1826,  time  of  first  payment, 

T  months  24  days, 13.649  + 

Amount,        .         .        .       $363.647 
Payment,  Dec.  25th,  exceeding  interest  then  due,    .          49.998 

Remainder  for  a  new  principal,        ....       $313.649 
Interest  of  $313.649,  from  Dec.  25th,  1826,  to  June 

14th,  1829,  2  years  5  months  19  days,  .         .          46.4721 

Amount,        .        .        .       $360.1211 
Payment,  July  10th,  1827,  less  than  in-j 
terest  then  due,  } 

Payment,  Sept.  21st,  1828,      .         .         .         15.008 

Their  sum,  less  than  interest  then  due,      $20.006 
Payment,  June  14th,  1829,      .         .         .         99.999 

Their  sum  exceeds  the  interest  then  due,     .        .        120.005 

Remainder  for  a  new  principal,  June  14th,  1829,    .       $240.1161 
Interest   of  $240.116,   from    June   14th,   1829,   to 

April  15th,  1830,  10  months  1  day,  .  .  12.0458 

Total  due,  April  15th,  1830,          .       $252.1619  + 


$3469.32 


NEW  YORK,  Feb.  6th,  1825. 


2.  For  value  received,  I  promise  to  pay  William  Jenks,  or 
order,  three  thousand  four  hundred  and  sixty-nine  dollars  and 
thirty-two  cents,  with  interest  from  date,  at  6  per  cent.  ? 

BILL  SPENDTHRIFT. 
On  this  note  were  indorsed  the  following  payments  : 

May  16th,  1828,  received  $545.76. 
May  16th,  1830,  .  .  1276.00. 
Feb.  1st,  1831,  .  .  2074.72. 

What  remained  due,  Aug.  llth,  1832? 

3.  A's  note,  of  $635.84,  was  dated  Sept.  5th,  1817;   on 
which  were  indorsed  the  following  payments,  viz. :  Nov.  13th, 
1819,  $416.08;    May  10th,   1820,   $152.00:    what 

March  1st,  1821,  the  interest  being  6  per  cent.?^f 

n 


244  PERCENTAGE. 


PROBLEMS    IN    INTEREST. 

297.  In  all  questions  of  Interest,  there  are  four  things 
considered,  viz.  : 

1st,  The  Principal;  2d,  The  Rate  of  Interest;  3d,  The 
Time ;  and  4th,  The  Amount  of  Interest. 

If  three  of  these  are  known,  the  fourth  can  be  found.  By 
Art.  292,  the  interest  is  found  by  multiplying  together  the 
principal,  rate,  and  time  in  years  ;  therefore,  the  interest  is 
the  product  of  the  principal,  rate,  and  time.  Either  of  these 
factors  is  found  by  dividing  the  product  by  the  product  of 
the  other  two  :  Hence,  we  have  the  following  principles : 

1st,  The  principal  is  equal  to  interest  divided  by  the  rate 
and  time:  2d,  The  rate  is  equal  to  interest  divided  by  the 
principal  and  time:  3d,  The  time  is  equal  to  interest,  di- 
vided by  principal  and  rate. 

Examples. 

1.  The  interest  of  a  certain  sum  for  4  years,  at  7  per  cent., 
is  $266  :  what  is  the  principal  ? 

OPERATION. 

ANALYSIS. — The  interest,  $266,  divided  0^x4  —  og 

by  the  product  of  the   rate   and  time,  *  .  ~  !.'__   * 

.07  x  4  =  .28.  will  give  the  principal.        2bb  **  '**  ~ 

principal. 

2.  The  interest  of  $3675,  for  3  years,  is  $171.75 :  what  is 
the  rate  ? 

3.  The  principal   is   $459,  the   interest   $183.60,  and  the 
rate  8  per  cent. :  what  is  the  time  ? 

4.  The  interest  of  a  certain  sum  for  3  years,  at  6  per  cent., 
is  $40.50  :  what  is  the  principal  ? 

5.  The  principal  is   $918,  the   interest   $269.28,   and  the 
rate  4  per  cent. :   what  is  the  time  ? 

297.  How  many  things  are  considered  in  every  question  of  in- 
terest ?  What  are  they  ?  The  interest  is  the  product  of  what  fac- 
tors ?  How  may  one  of  these  factors  be  found  ?  What  are  the  three 
principles  ? 


COMPOUND   INTEREST.  245 

6.  What  sum  of  money  must  be  placed  on  interest  at  7  per 
cent.,  for  3  yrs.  9  mos.,  that  the  interest  may  be  $396  ? 

7.  In    what    time,    at    7    per   cent.,   will    a    mortgage    of 
$8762.50,  whose  interest  is  unpaid,  amount  to  $10000? 

8.  If,  by  purchasing  a  house  for  $5620,  I  have  received, 
in  2  yrs.  3  mos.  15  days,  $1800  rent:    what  rate  of  interest 
have  I  received  ?| 

9.  A  merchant,  who  had  bought  goods  for  $15960,  sold 
them,  at  the  end  of  5  months  16  days,  at  an  advance  of  27 
per  cent. :  what  rate  of  interest  did  he  receive  ? 

10.  What  sum  of  money,  at  6  per  cent.,  will  produce,  in 
2  yrs.  9  mos.  10  days,  the  same  interest  that  $350  produces, 
at  8  per  cent.,  in  3  yrs.  10  mos.  5  days  ? 

11.  In  what  time  will  $5000,  at  7  per  cent.,  produce  the 
same  interest  that  $9625  produces,  at  6|  per  cent.,  in  4  yrs. 
5  mos.  18  days? 


COMPOUND  INTEREST. 

298.  COMPOUND  INTEREST  is  the  interest  of  the  amount  of 
the  principal  and  its  unpaid  interest. 

This  interest  may  be  computed  annually,  semi-annually, 
quarterly,  monthly,  or  daily*  In  Savings  Banks,  the  interest 
is  generally  computed  semi-annually. 

From  the  definition,  we  deduce  the  following 

Rule. —  Compute  the  interest  to  the  time  at  which  it 
becomes  due;  then  add  it  to  the  principal,  and  compute 
the  interest  on  the  amount  as  on  a  new  principal :  add  the 
interest  again  to  the  principal,  and  compute  the  interest  as 
before;  do  the  same  for  all  the  times  at  which  payments  of 
interest  become  due ;  from  the  last  result  subtract  the  prin- 
cipal, and  the  remainder  will  be  the  compound  interest. 

208.  What  of  Compound  Interest  ?    How  do  you  compute  it  ? 


246 


PERCENTAGE. 


Examples. 

1.  What  will  be  the  compound  interest,  at  7  per  cent.,  of 
$3750,  for  2  years,  the  interest  being  added  yearly  ? 

OPERATION. 

3750 
.07 


262.50,   Interest  for  1st  year. 
3750 

4012.5a,   Principal  for  2d  year. 
,07 


280.8750,   Interest  for  2d  year. 
4012,50 


4293.375, 
3750 

$543.375, 


Amount  at  2  years. 
1st  Principal. 

Compound  interest. 


NoTE.-4-When  there  are  months  and  (jlays  in  the  time,  find  the 
amount  wr  the  years,-  and  on  this  ainojimt  cast  the  -interest  "for 
the  months  and  days  :  this,  added  to  thp  last  amount,  will  be  the 
required  amount  for  the  whole 


2.  If  the  interest  be  computed  annually,  what  will  be  the 
compound  interest  on  $100,  for  3  years,  at  6  per  cent.  ? 

3.  What  will  be  the  compound  interest  on  $295.37.  at  6 
per  cent.,  for  2  years,  the  interest  being  added  annually? 

4.  What  will  be  the  compound  interest,  at  5  per  cent.,  of 
$1875,  for  4  years  ? 

5.  What  is  the  amount,  at  compound  interest,  of  $250, 
for  2  years,  at  8  per  cent.  ? 

6.  What  is  the  compound  interest   of  $939.64,  for  3  yr. 
9  mo.,  at  7  per  cent.  ? 

7.  What  will  $125.50   amount  to,  in  10  years,  at  4  per 
cent,  compound  interest  ? 

>  8.  What  will  be  the  amount  of  $250,  which  has  been  in 
savings  bank  for  2  years,  supposing  interest  computed  semi- 
annually,  at  6  per  cent.  ?  What  the  compound  interest  ? 


COMPOUND    INTEREST. 


247 


9.  What  will  be  the  interest  of  $500,  which  has  been  in 
savings  bank  for  1  yr.  6  mo.,  supposing  the  interest  to  be 
computed  semi-annually,  at  5  per  cent.? 

NOTE. — The  operation  is  rendered  much  shorter  and  easier,  by 
taking  the  amount  of  1  dollar  for  any  time  and  rate,  given  in  the 
following  table,  and  multiplying  it  by  the  given  principal ;  the 
product  will  be  the  required  amount,  from  which  subtract  the 
given  principal,  and  the  result  will  be  the  compound  interest. 

The  result  may  differ  in  the  mills  place,  from  that  obtained  by 
the  other  rule. 

Table, 

Showing  the  amount  of  $1  or  £1,  compound  interest,  from  1  year 
to  20,  and  at  the  rate  of  3,  4,  5,  6,  and  7  per  cent. 


Years. 

8  per  cent 

4  per  eent 

5  per  cent 

6  per  cent 

7  per  cent 

1 

1.03000 

1.04000 

1.05000 

1.06000 

1.07000 

2 

1.06090 

1.18160 

1.10250 

1.12360 

1.14490 

3 

1.09272 

1.12486 

1.15762 

1.19101 

1.22504 

4 

1.12550 

1.16985 

1.21550 

1.26247 

1.31079 

5 

1.15927 

1.21665 

1.27628 

1.33822 

1.40255 

6 

1.19405 

1.26531 

1.34009 

1.41851 

1.50073 

7 

1.22987 

1.31593 

1.40710 

1.50363 

1.60578 

8 

1.26677 

1.36856 

1.47745 

1.59384 

1.71818 

9 

1.30477 

1.42331 

1.55132 

1.68947 

1.83845 

10 

1.34391 

1.48028 

1.62889 

1.79084 

1.96715 

11 

1.38423 

1.53945 

1.71033 

1.89829 

2.10485 

12 

1.42576 

1.60103 

1.79585 

2.01219 

2.25219 

13 

1.46853 

1.66507 

1.88564 

2.13292 

2.40984 

14 

1.51258 

1.73167 

1.97993 

2.26090 

2.57853 

15 

1.55796 

1.80094 

2.07892 

2.39655 

2.75903 

16 

1.60470 

1.87298 

2.18287 

2.54035 

2.95216 

17 

1.65284 

1.94790 

2.29201 

2.69277 

3.15881 

18 

1.70243 

2.02581 

2.40661 

2.85433 

3.37993 

19 

1.75350 

2.10684 

2.52695 

3.02559 

3.61652 

20 

1.80611 

2.19112 

2.65329 

3.20713 

3.86968 

10.  What  is  the  amount  of  $96.50,  for  8  years  and  6  mo., 
interest  being  compounded  annually,  at  7  per  cent.? 

11.  What  is  the  compound  interest  of  $300,  for  5  years, 
8  months,  and  15  days,  at  6  per  cent.  ? 

12.  What  is  the  compound  interest  of  $1250,  for  3  years, 
3  months,  and  24  days,  at  7  per  rent.  ? 


248 


PERCENTAGE. 


DISCOUNT. 

299.  DISCOUNT  is  an  allowance  made  for  the  payment  of 
Kmey  before  it  is  due. 

300.  PRESENT  VALUE  of  a  debt,  due  at  a  future  time,  is 
such  a  sum  as,  being  placed  at  interest  until  the  debt  becomes 
due,  would  increase  to  an  amount  equal  to  the  debt. 

301.  FACE  of-  a  note,  is  the  amount  named  in  a  note. 

302.  DISCOUNT   on  a  note,   is   the   difference  between  the 
face  of  a  note  and  its  present  value. 

1.  I  give  my  note  to  Mr.  Wilson,  for  $107,  payable  in  1 
year:  what  is  the  present  value  of  the  note,  if  the  interest 
is  7  per  cent.  ?  What  the  discount  ? 

ANALYSIS. — Since  $1  in  1  year,  at  7  per  cent.,  will  amount  to 
$1.07,  the  present  value  of  $1.07  is  $1 :  hence, 

1  :  1+-07  ::  present  val.  of  any  sum  :  face  of  note; 

face  of  note       TT 
or,  Present  val.  =  — -- —  ;     Hence, 

Rule.— \Divide  the  face  of  the  note  by  1  dollar  plus  the 
interest  of  1  dollar  for  the  given  time,  and  the  quotient  will 
be  the  present  value :  take  this  sum  from  the  face  of  the 
note,  and  the  remainder  will  be  the  discount. 

Examples. 

1.  What  is  the  present  value  of  a  note  for  $1828.75,  due 
1  year,  and  bearing  an  interest  of  4J  per  cent.  ? 

2.  A  note  of  $1651.50  is  due  in  11  months,  but  the  per- 
son to  whom  it  is  payable,  sells  it  with  the  discount  off,  at 
6  per  cent. :   how  much  shall  he  receive  ? 

NOTE. — When  payments  are  to  be  made  at  different  times, 
find  the  present  value  of  the  sums  separately,  and  their  sum  will 
l)e  the  present  value  of  the  note. 


DISCOUNT.  249 

3.  What  is   the  present  value   of  a  note  for  $10500,  on 
which  $900  are  to  be  paid  in  6  months,  $2700  in  one  year, 
$3900  in  eighteen  months,  and  the  residue  at  the  expiration 
of  2  years,  the  rate  of  interest  being  6  per  cent,  per  annum  ? 

4.  What  is  the  discount  of  $4500,  one-half  payable  in  six 
months,  and  the  other  half  at  the  expiration  of  a  year,  at  7 
per  cent,  per  annum  ? 

5.  What  is  the  present  value  of  $5760,  one-half  payable 
in  3  months,  one-third  in  6  months,  and  the  rest  in  9  months, 
at  6  per  cent,  per  annum  ? 

6.  Mr.  A  gives  his  note  to  B,  for  $720,  one-half  payable 
in  4  months,  and  the  other  half  in  8  months  :   what  is  the 
present  value  of  said  note,  discount  at  5  per  cent,  per  annum  ? 

7.  What  is  the  difference  between  the  interest  and  discount 
of  $750,  due  nine  months  hence,  at  7  per  cent.  ? 

8.  What  is  the  present  value  of  $4000,  payable  in  9  months, 
discount  4 1  per  cent,  per  annum  ? 

9.  Mr.   Johnson    has    a    note    against    Mr.  Williams,    for 
$2146.50,  dated  August  17th,  1862,  which  becomes  due  Jan. 
llth,  1863  :    if  the  note  is  discounted  at  6  per  cent.,  what 
ready  money  must  be  paid  for  it  September  25th,  1862? 

10.  C  owes  D  $3456,  to  be  paid  October  27th,  1862  ;  C 
wishes   to   pay   on  the   24th   of  August,  1858,   to  which  D 
consents :   how  much  ought  D  to  receive,  interest  at  6  per 
cent.  ? 

11.  What  is  the  present  value  of  a  note  of  $4800,  due  4 
years  hence,  the  interest  being  computed  at  5  per  cent,  per 
annum  ? 

12.  A  man  having  a  horse  for  sale,  offered  it  for  $985 
cash  in  hand,   or   $230  at   9  months ;   the  buyer  chose   the 
latter:    did  the   seller  lose   or  make  by  his  offer,  supposing 
money  to  be  worth  7  per  cent.  ? 

299.  What  is  Discount  ?— 300.  What  is  present  value  of  a  debt  ? 

801.  What  is  face  of  a  note? 

802.  What  is  discount  on  a  note  ?    What  is  the  rule  for  finding 
the  present  value  ?    What  is  the  discount  ? 


250 


PERCENTAGE. 


BANKING-. 

303.  A  BANK  is  a  joint  stock  company,  incorporated  for 
the  purpose  of  loaning  money  and  issuing  bills  for  currency. 

304.  BANK  DISCOUNT  is  the  charge  made  by  a  bank,  for 
the  use  of  money  payable  at  a  future  time. 

305.  A  PROMISSORY  NOTE  is  a  written  obligation  to  pay  a 
specified  sum  at  a  future  time,  named  in  the  note. 

By  the  custom  of  banks,  the  interest  is   always  paid  in 
advance. 

306.  By  mercantile  usage,  a  note  at  bank  does  not  legally 
fall  due  until  3  days  after  the  expiration  of  the  time  named 
on  its  face;  these  days  are  called,  Days  of  Grace. 

The  present  value  of  a  note,  is  sometimes  called  its  Cash 
Yalue,  or  Proceeds,  or  Avails. 


CASE     I. 
307.    To  find  the  bank  discount  and  present  worth. 

Rule. — Add  the  3  days  of  grace  to  the  time  the  Note 
has  to  run,  and  then  calculate  the  interest  at  the  given  rate: 
the  result  is  the  bank  discount,  and  the  face  of  the  'note 
diminished  by  the  discount,  is  the  present  worth. 

Examples. 

1.  What  is  the  bank  discount  on  a  note  for  $350,  payable 
3  months  after  date,  at  7  per  cent,  interest  ? 

2.  What  is  the  bank  discount  of  a  note  of  $1000,  payable 
in  60  days,  at  6  per  cent,  interest  ? 

3.  A  merchant   sold   a   cargo   of  cotton   for  $15720,  for 
which  he  receives  a  note  at  6  months  :  how  much  money  will 
he  receive  at  a  bank  for  this   note,  discounting  it  at  6  per 
cent,  interest  ? 


BANKING.  251 

4.  What  is  the  bank  discount  on  a  note  of  $556.27,  pay- 
able in  60  days,  discounted  at  6  per  cent,  interest  ? 

5.  A  has  a  note  against  B,  for  $3456,  payable  in  three 
months  ;  he  gets  it  discounted,  at  7  per  cent,  interest :  how 
much  does  he  receive  ? 

6.  What  is  the  bank  discount  on  a  note  of  $367.47,  hav- 
ing 1  year,  1  month,  and  13  days  to  run,  as  shown  by  the 
face  of  the  note,  discounted  at  7  per  cent.? 

7.  What   is   the  bank  discount,  on  a  note  at  5  months, 
for  $672.50,   dated  August  7th,  1862,  the  interest  being  6 
per  cent.? 

8.  Mr.  Jones  gave  his  note  at  bank,  for  8  months,  at  7  per 
cent.,  for  $1670  :  what  was  the  product? 

NEW  YORK,  July  3d,  1860. 

9.  For  value  received,  I  promise  to  pay  to  John  Jones,  on 
the  20th  of  November  next,  six  thousand  five  hundred  and 
seventy-nine  dollars  and  fifteen  cents. 

PHILIP  MASON. 

What  will  be  the  discount  on  this,  if  discounted  on  the  1st 
of  August,  at  6  per  cent,  per  annum  ? 

CASE    II. 

308.  To  make  a  note  due  at  a  future  time,  whose  present 
value  shall  be  a  given  amount. 

1.  For  what  sum  must  a  note  be  drawn  at  3  months,  so 
that,  when  discounted  at  a  bank  at  6  per  cent.,  the  amount 
received  shall  be  $500  ? 

ANALYSIS. — If  we  find  the  interest  on  1  dollar  for  the  given 
time,  and  then  subtract  that  interest  from  1  dollar,  the  Remainder 

303.  What  is  a  bank  ?— 304.  What  is  bank  discount  ?— 305.  What 
is  a  promissory  note  ?  How  is  the  interest  paid  at  a  bank  ? 

306.  By  mercantile  usage,  when  does  a  note  at  bank  fall :  due? 
What  are  these  days  called? 

307.  How  do  you  find  the  discount  and  present  worth? 

308.  How  do  you  make  a  note  payable  at  a  future  time,  whose 
present  value  shall  be  a  given  amount  ? 


252  PERCENTAGE. 

will  be  the  present  value  of  1  dollar,  due  at  the  expiration  of  that 
time.  Then,  the  number  of  times  which  the  present  value  of  the 
note  contains  the  present  value  of  1  dollar,  will  be  the  number 
of  dollars  for  which  the  note  must  be  drawn. 

OPERATION. 

Interest  of  $1  for  the  time,  3  mos.  and  3  days  —  $0.0155,  which 
taken  from  $1,  gives  present  value  of  $1=0.9845;  then,  $500-h 
0.9845  =  $507.872+  =face  of  note. 

PKOOF. — Bank  interest  on  $507.872,  for  3  months,  including  3 
days  of  grace,  at  6  per  cent.  =  7.872 ;  which  being  taken  from 
the  face  of  the  note,  leaves  $500  for  its  present  value. 

Hence,  we  have  the  following 

Rule. 

Divide  the  present  value  of  the  note  by  the  present  value 
of  1  dollar,  reckoned  for  the  same  time  and  at  the  same  rate 
of  interest,  and  the  quotient  will  be  the  face  of  the  note. 

Examples. 

1.  For  what  sum  must  a  note  be  drawn,  at  7  per  cent., 
payable  on  its  face  in  1  year  6  months  and  15  days,  so  that, 
when  discounted  at  bank,  it  shall  produce  $307.27  ? 

2.  A  note  is  to  be  drawn,  having  on  its  face  8  months  and 
12  days  to  run,  and  to  bear  an  interest  of  7  per  cent.,  so  that 
it  will  pay  a  debt  of  $5450  :  what  is  the  amount  ? 

3.  What  sum,  6  months  and  9  days  from  July  18th,  1862, 
drawing  an  interest  of  6  per  cent.,  will  pay  a  debt  of  $674.89 
at  bank,  on  the  1st  of  August,  1862  ? 

4.  Mr.  Jones  bought  a  bill  of  goods  amounting  to  $1683.75, 
and  paid  the  bill  with  a  note  running  6  months  :    for  what 
amount  must  the  note  be  drawn,  that,  when  discounted,  the 
merchant  will  receive   exactly  the  amount   of  the  bill,   sup- 
posing interest  to  be  at  7  per  cent.? 

5.  Mr.  Wilson   is   indebted   at   the   bank   in   the   sum   of 
$367.464,  which  he  wishes  to  pay  by  a  note  at  4  months, 
with  interest  at  7  per  cent". :  for  what  amount  must  the  note 
be  drawn  ? 


EXCHANGE.  253 


EXCHANGE. 

309.  EXCHANGE  is  a  process  of  remitting  money  from  one 
place  to  another,  by  means  of  written  orders. 

310.  A  BILL  OF  EXCHANGE  is  a  written  order  from   one 
person  to  another,  to  pay  to  a  third  party  a  specific  sum,  at 
a  given  time. 

311.  The  DRAWER,   or  MAKER,  is  the  person  who   draws 
the  bill. 

312.  The  DRAWEE  is  the  person  on  whom  the  bill  is  drawn. 

313.  The  PAYEE  is  the  person  to  whom  the  money  is  or- 
dered to  be  paid ;  and  he  is  the  owner  of  the  bill. 

314.  The  PAYER,  or  REMITTER,  is  any  person  who  purchases 
a  bill  of  exchange,  either  of  the  drawer  or  payee. 

315.  An  ACCEPTANCE  is  an  agreement  of  the  drawee,  to 
pay  the  bill  when  it   falls  due,   and   is   signified  by  writing, 
"Accepted,"  with  his  signature,  on  the  face  of  the  bill. 

316.  An   INDORSEMENT   of   a   bill,    by   the   payee,    is    the 
writing  of  his  name  on  the  back  of  it.     This  transfers  the 
bill  to   any  person  who   may  rightfully  hold   it.     Or,  if  he 
writes   on  the  back,  "  Pay  to  John  James  or  order,"  then 
the  bill  is  transferred  to  Mr.  James. 

317.  DAYS  OF  GRACE  are  days  granted  to  the  person  who 
pays  a  bill,  after  the  time  named  in  it  has  expired.     In  the 
United  States  and  Great  Britain,  3  days  are  generally  allowed. 

318.  An  INLAND   BILL  is   when   the   drawer  and   drawee 
both  reside  in  the  same  country. 

319.  The  COURSE  OF  EXCHANGE   is  the  difference  between 
the  face  of  a  bill,  and  the  price  paid  for  it. 

320.  PAR  OF  EXCHANGE  is  when  the  face  of  a  bill  and  the 
price  paid,  are  the  same. 


254  PERCENTAGE. 

321.  PREMIUM  is  when  the  price  paid  is  greater  than  the 
face  of  the  bill,  and  the  exchange  is  then  said  to  be,  above  par 

322.  DISCOUNT  is  when  the  price  paid  is  less  than  the  face 
of  the  bill,  and  the  exchange  is  then  said  to  be,  below  par. 

Rules  of  Articles  251,  262,  and  308,  apply  to  all  cases  of 
Exchange. 

Examples. 

1.  A  merchant   at  Chicago  wishes  to  pay  a  bill  in  New 
York,  amounting  to  $3675,  and  finds  that  exchange  is   1J 
per  cent,  premium :   what  must  he  pay  for  his  bill  ? 

2.  A  merchant  in  Philadelphia  wishes  to  remit  to  Charles- 
ton, $8756.50,  and  finds   exchange  to  be  1  per  cent,  below 
par :   what  must  he  pay  for  the  bill  ? 

3.  A   merchant  in  Mobile  wishes  to  pay  in  New  York, 
$6584,   and  exchange  is   2|    per  cent,  premium:   how  much 
must  he  pay  for  such  a  bill  ? 

4.  A  merchant  in  Boston  wishes  to  pay  in  New  Orleans, 
$4653.75  ;  exchange  between  Boston  and  New  Orleans  is  1| 
per  cent,  below  par  :  what  must  he  pay  for  a  bill  ? 

5.  A  merchant  in  New  York  has  $3690,  which  he  wishes 
to  remit  to  Cincinnati;  the  exchange  is  1|  per  cent,  below 
par :  what  will  be  the  amount  of  his  bill  ? 

6.  What  must  be  paid  for  the  following  bill,  when  exchange 
is  at  2£  per  cent,  premium? 


$950.50.  NEW  YORK,  May  9th,  1860. 

Thirty  days  after  date,  pay  to  Samuel  Lewis  or  order, 
Nine  hundred  and  Fifty  dollars  and  Fifty  cents,  and  charge 
the  same  to  my  account.  DANIEL  SHAW. 

To  Messrs.  JONES  &  WILLIS. 

7.  If  exchange  is  at  1J  per  cent,  premium,  what  bill  will 
$3950  purchase? 

8.  If  exchange  is   at   1|  per  cent,  premium,  what  bill  of 
exchange  can  be  bought  for  $762,  uncurrent  funds,  supposing 
a  discount  of  \  per  cent,  is  charged  on  the  funds  ? 


EXCHANGE.  255 


FOREIGN    BILLS. 

323.  A  FOIIEIGN  BILL  OF  EXCHANGE  is  one  in  which  the 
drawer  and  drawee  live  in  different  countries. 

England  and  France  are  the  principal  countries  with  which 
the  United  States  have  exchanges. 

In  all  Bills  of  Exchange  on  England,  the  £  sterling  is 
the  unit  or  base,  and  is  still  reckoned  at  its  former  value  of 
$4£  =  $4.4444+,  instead  of  its  present  value,  $4.84. 

Hence,      .....    £1  =  $4.4444  + 
Add  9  per  cent,    ....  .3999 

Gives  the  present  value  of  £1,          $4.8443. 

Hence,  the  true  par  value  of  Exchange  on  England,  is  9 
per  cent,  on  the  nominal  base. 

324.  To  find  the  value  of  a  bill  of  exchange  in  sterling 
money,  in  United  States  money. 

1.  A  merchant  in  New  York,  wishes  to  remit  to  England, 
a  bill  of  exchange  for  £125  15s.  6d.  :  how  much  must  he  pay 
for  this  bill,  when  exchange  is  at  9|  per  cent,  premium  ? 

£125  15s.  6d  ......   =  £125.775 

Add  9£  per  cent  .....  11.9486  + 

gives  amount  in  £'s,        .       .       .        £137.7236  + 

The  pounds  and  decimals  of  a  pound  are  reduced  to  dollars, 
by  multiplying  by  40  and  dividing  by  9,  —  giving,  in  this  case, 
$612.105  (Art. 


Rule.  —  I.  Reduce  the  amount  of  the  bill  to  pounds  and 
decimals  of  a  pound,  and  then  add  the  premium  of  exchange. 

II.  Multiply  the  result  by  40,  and  divide  the  product  by 
9  :  the  quotient  will  be  the  answer  in  United  States  money. 

Examples. 

1.  A  merchant  shipped  100  bales  of  cotton  to  Liverpool, 
each  weighing  450  pounds.  They  were  sold  at  7Jd.  per  pound, 
and  the  freight  and  charges  amounted  to  £187  10s.  He 


256  PERCENTAGE. 

sold  his  bill  of  exchange  at  9|  per  cent,  premium :  how  much 
should  he  receive  in  United  States  money  ? 

2.  There  were  shipped  from  Norfolk,  Va.,  to  Liverpool, 
85  hhd.  of  tobacco,  each  weighing  450  pounds.  It  was  sold 
at  Liverpool,  for  12|d.  per  pound,  and  the  expenses  of  freight 
and  commissions  were  <£92  Is.  8d.  If  exchange  on  New  York 
is  at  a  premium  of  9J  per  cent.,  what  should  the  owner 
receive  for  the  bill  of  exchange,  in  United  States  money  ? 

325.    Exchange  on  France. 

The  unit  or  base  of  the  French  Currency,  is  the  French 
franc,  of  the  value  of  18  cents  6  mills.  The  franc  is  divided 
into  tenths,  called  decimes,  corresponding  to  our  dimes,  and 
into  centimes,  corresponding  to  cents.  Thus,  5.12  is  read, 
5  francs  and  12  centimes. 

All  Bills  of  Exchange  on  France,  are  drawn  in  francs. 
Exchange  is  quoted  in  New  York,  at  so  many  francs  and 
centimes  to  the  dollar. 

1.  What  will  be  the  value  of  a  bill  of  exchange  for  4536 
francs,  at  5.25  to  the  dollar? 

ANALYSIS. — Since  1  dollar  will  buy  OPERATION. 

5.25  francs,  the  bill  will  cost  as  many       5.25  )  4536  (  $864j  Ans. 
dollars  as  5.25  is  contained  times  in 
the  amount  of  the  bill :     Hence, 

Rule. — Divide  the  amount  of  the  bill  by  the  value  of  $1 
in  francs :  the  quotient  is  the  amount  to  be  paid  in  dollars. 

Examples. 

1.  What  will  be   the   amount   to   be  paid,  United  States 
money,  for  a  bill  of  exchange  on  Paris,  of  6530  francs,  ex- 
change being  5.14  francs  per  dollar? 

2.  What  will  be  the  amount  to  be  paid,  in  United  States 
money,  for  a  bill  of  exchange  on  Paris,  of  10262  francs,  ex- 
change being  5.09  francs  per  dollar  ? 

3.  What  will  be  the  value,  in  United  States  money,  of  a 
bill  for  87595  francs,  at  5.16  francs  per  dollar? 


EQUATION  OF  PAYMENTS.  257 


EQUATION    OF    PAYMENTS. 

326.  EQUATION  OF  PAYMENTS  is  the  operation  of  finding 
the  time  in  which  several  sums,  due  at  different  times,  may 
be  paid  without  loss  of  interest  to  either  party.     The  time 
of  payment  thus  found  is  called  the  mean  time,  or  equated 
time. 

327.  When  the  times  of  payment  are  reckoned  from  the 
same  date. 

1.  If  I  owe  Mr.  Wilson  $2,  to  be  paid  in  6  months,  from 
July  1st  ;  $3,  to  be  paid  in  8  months  ;  and  $1,  to  be  paid  in 
12  months  :  what  is  the  mean  time  of  payment  ? 

ANALYSIS.  —  The  interest  on  all  the  sums,  to  their  various  times 
of  payment  equals  the  interest  of  $48  for  one  month  ;  but  $48  is 
equal  to  the  sum  of  all  the  products  which  arise  from  multiplying 
each  sum  by  the  time  at  which  it  becomes  due.  It  will  take  $6 
(the  sum  of  the  payments)  as  many  months,  to  produce  the  same 
interest,  as  $6  is  contained  times  in  $48,  which  is  8  times  :  there- 
fore, the  equated  time  is  8  months. 

OPERATION. 

Int.  of  $2  for  6  mos.  =  int.  of  $12  for  1  mo.    2x    6  —  12. 

"       $3    "    8     "     =  int.  of  $24    "  "        3x8  =  24. 

"       $1    "12     "     =  int.  of  $12    "  "       J_xl2  =  12. 

$6                                      $48  6 


Rule. 

Multiply  each  payment  by  the  time  before  it  becomes  due, 
and  divide  the  sum  of  the  products  by  the  sum  of  the  pay- 
ments: the  quotient  will  be  the  mean  time. 

Examples. 

1.  A  merchant  owes  $600,  to  be  paid  in  12  months  from 
January  1st  ;  $800,  to  be  paid  in  6  months,  and  $900,  to  be 
paid  in  9  months  :  what  is  the  equated  time  of  payment  ? 


25S 


EQUATION    OF   PAYMENTS. 


2.  A  owes  B  $600 ;   one-third  is  to  be  paid  in  6  months 
from  August  1st ;  one-fourth  in  8  months,  and  the  remainder 
in  1 2  months :  what  is  the  mean  time  of  payment  ? 

3.  A  merchant  has  due  him  $300,  to  be  paid  in  60  days, 
$500,  to  be  paid  in  120  days,  and  $750,  to  be  paid  in  180 
days  :  what  is  the  equated  time  of  payment  ? 

328.   When  the  times  are  reckoned  from  different  dates. 

1.  I  owe  Mr.  Wilson  $100,  to  be  paid  on  the  15th  of 
July ;  $200,  on  the  15th  of  August ;  and  $300,  on  the  9th 
of  September :  what  is  the  mean  time  of  payment  ? 

ANALYSIS. — The  earliest  date  named,  or  any  day  previous  to  it, 
may  be  taken  as  the  date  from  which  the  times  are  reckoned. 
If  the  earliest  payment  is  the  date  of  reckoning,  the  first  multi- 
plier is  0,  and  consequently  the  first  product  is  0,  but  the  pay- 
ment must  be  added,  in  finding  the  sum  of  the  payments. 

OPERATION. 

From  1st  of  July  to  1st  payment,  14  days. 
"        "         "        to  2d  payment,  45  days. 
"         "         to  3d  payment,  70  days. 
100  x  14  =    1400 
200  x  45  =    9000 
300  x  70  =  21000 

600        6100  )  314|00 


Hence,  the  equated  time  is  52^  days  from  the  1st  of  July ; 
that  is,  on  the  23d  day  of  August. 

But  if  we  estimate  the  time  from  the  15th  of  July,  we 
shall  have, 

From  July  15th  to  1st  payment,     0  days. 
"         "        "      to  2d  payment,  31  days. 
"         "        "      to  3d  payment,  56  days. 
Then,  100  x     0  =      000 

X  31   =    6200 
X  56  '  =  16800 


200 
300 

600 


6|00  )  230|00 
38J 


EQUATION  OF  PAYMENTS.  259 

Hence,  the  payment  is  due  in  38J  days  from  July  15th ;  or, 
on  the  23d  of  August — the  same  as  before. 

Rule. — Assume  the  earliest  date  as  the  point  of  reck- 
oning. Find  the  number  of  days  intervening  between  this 
date  and  that  of  each  payment,  and  multiply  each  sum. 
by  its  number  of  days:  add  the  products,  and  divide  the 
sum  by  the  sum  of  the  amounts,  and  the  quotient  will  be 
the  equated  time  in  days.  This  number,  reckoned  from 
the  earliest  date,  will  give  the  equated  date. 

Examples. 

1.  I  owe  $1000,  to  be  paid  on  the  1st  of  January ;  $1500, 
on  the  1st  of  February;   $3000,  on  the  1st  of  March;  and 
$4000,  on  the  15th  of  April:  reckoning  from  the  1st  of  Jan- 
uary, and  calling  February  28  days,  on  what  day  must  the 
money  be  paid  ? 

NOTE.— If  one  of  the  payments,  as  in  the  above  example,  is 
due  on  the  day  from  which  the  equated  time  is  reckoned,  its 
corresponding  product  will  be  nothing,  but  the  payment  must 
still  be  added  in  finding  the  sum  of  the  payments. 

2.  Mr.  Jones  purchased  of  Mr.  Wilson,  on  a  credit  of  sixty 
days,  goods  to  the  following  amounts : 

15th  of  January,  a  bill  of  $3750. 

10th  of  February,  a  bill  of    3000. 

6th  of  March,  a  bill  of    2400. 

8th  of  June,  a  bill  of    2250. 

He  wishes,  on  the  first  of  July,  to  give  his  note  for  the 
amount :  at  what  time  must  it  be  made  payable  ? 

3.  A  merchant  bought  several  lots  of  goods,  as  follows : 

A  bill  of  $650,  June  6th. 
A  bill  of  890,  July  8th. 
A  bill  of  7940,  Aug.  1st. 

Now,  if  the  credit  is  6  months,  how  many  days  from  De- 
cember 6th  before  the  note  becomes  due  ?  At  what  time  ? 


260 


EQUATION   OF   PAYMENTS. 


1861. 


G.  DUCK, 
Jan.        2d. 
Jan.      31st. 
Feb.       5th. 
March  19th. 


4.  Mr.  Tappan  rendered  to  Mr.  Duck  the  following  bill  of 

goods  sold  him: 

NEW  YORK,  May  6th,  1861. 

To  A.  TAPPAN,     Dr. 
To  merchandise  on  4  mos.,     $1215 

1600 
«  »  «  595 

"  "  "  675 

What  is  the  equated  time  of  payment  of  this  bill  ? 

5.  A  merchant  owes  the  following  bill : 

1860.  May  6th.  To  merchandise  on  6  mos.,  $  892 
June  19th.  "  "  "  1063 

July  4th.  "  "  "  2585 

What  is  the  equated  time  of  payment  from  July  4th  ? 

328.  To  settle,  by  payment  of  cash,  an  account  in  •which 
there  are  debtor  and  creditor  items. 

BALANCE  is  a  term  which  denotes  the  difference  between 
the  debtor  and  creditor  sides  of  an  account.  There  are  three 
balances  ;  viz.,  merchandise  balance,  interest  balance,  and  cash 
or  net  balance. 

1.  MERCHANDISE  BALANCE  is  the  balance,  in  which  interest 
on  the  items  is  not  considered. 

2.  INTEREST  BALANCE  is  the  balance  of  the  interest  of  the 
items  of  the  two  sides. 

3.  CASH  or  NET  BALANCE  is  the  balance  which  arises  after 
adding  the  merchandise  and  interest  balances  to  the  proper 
sides  of  the  account. 

In  equating  the  cash  balance,  interest  is  allowed  on  each 
item,  and  the  balance  of  interest  becomes  a  new  item,  and 
must  be  added  to  its  proper  side  of  the  account. 

1.  It  is  required  to  find  the  cash  balance  of  the  following 
account,  on  April  1st,  1861. 
Dr.  ROBERT  CHEAP.  Cr. 


1861 

Jan. 

7 

To  Merchandise 

$750 

00 

Feb. 

9 

By  Cash 

$560 

00 

" 

10 

"            " 

816 

00 

M'h 

10 

"  Merchandise 

175 

00 

Feb. 

14 

1C                              CC 

195 

00 

" 

27 

<(             « 

160 

00 

Apr. 

1 

"  Interest  Bal. 

19 

54 

Apr. 

1 

"  Cash  Bal. 

785 

54 

EQUATION   OF   PAYMENTS. 


261 


Interest  on  each  item  is  calculated  from  April  1st.  We  reckon 
the  number  of  days  from  April  1st  to  each  date,  and  these  are 
used  as  multipliers. 


Dr.  items. 

195  x  46  =  8970 
816  x  75  =  61200 
750  x  84  =  63000 

133170 


Cr.  items. 

560  x  51  =  28560 
175  x  16  =  2800 
260  x  5  =  1300 

32660 


The  difference  of  the  sums  of  these  products  is  multiplied  by 
the  interest  of  $1  for  one  day,  and  this  gives  the  interest  balance. 
The  difference  of  the  products  is  133170  —  32660  =  100510,  and 
the  interest  for  1  day  is  .07  -r-  360  ;  hence, 


100510  x         = 


° 


=  $19.54  =  balance  of  interest. 


This  balance  belongs  to  the  debtor  side,  because  the  sum  of 
the  debtor  products  is  the  greater.  It  is,  therefore,  added  to  the 
debtor  side,  and  the  final  balance  is  the  cash  or  net  balance. 

Rule.  —  I.  Take  the  latest  date  of  the  account,  or  any 
later  date,  at  which  the  balance  is  to  be  struck,  as  the  point 
of  reckoning,  and  find  the  days  between  this  date  and  the 
date  of  each  item;  and  consider  these  days  as  multipliers. 

II.  Multiply  each  item  by  its  multiplier;  then  take  the 
difference  of  the  sums  of  these  products,  and  multiply  it 
by  the  interest  of  $1  for  one  day  :  the  result  will  be  the 
interest  balance,  which  is  to  be  added  to  the  side  having 
the  greater  sum. 

III.  Then  find  the  difference  of  the  sums  in  the  two  col- 
umns, and  this  will  be  the  cash  balance. 

2.  What  was  the  cash  balance  on  Aug.  1st,  1862,  of  the 
following  account  ? 


Dr. 


RICHARD  MONEYPENNY. 


Cr. 


1862 

May 

16 
21 

To  Cash 
"  Merchandise 

$716 
595 

00 
00 

May 

1 

12 

By  Merchandise 

$975:00 
1640|00 

June 

19 

tt            fi 

1697 

75 

June 

17 

"  Cash 

500iOO 

July 

7 
13 

"  Cash 

950 
176 

00 
00 

July 

1 

"  Mdse. 

615 

00 

262  PERCENTAGE. 


ASSESSING    TAXES. 

329.  A  TAX  is  a  certain  sum  required  to  be  paid  by  the 
inhabitants   of  a  town,  county,  or  State,  for  the  support  of 
government.     It  is  generally  collected  from  each  individual  in 
proportion  to  the  amount  of  his  property. 

Property  is  of  two  kinds,  real  and  personal.  Real  prop- 
erty, or  real  estate,  is  fixed  property,  such  as  houses  and 
lands.  Personal  property  is  movable  property,  such  as  money, 
furniture,  &c. 

In  some  States,  however,  every  white  male  citizen  over  the 
age  of  twenty-one  years  is  required  to  pay  a  certain  tax. 
This  tax  is  called  a  poll-tax ;  and  each  person  so  taxed  is 
called  a  poll. 

330.  In  assessing  taxes,  the  first  thing  to  be  done  is  to 
make  a  complete  inventory  of  all  the  property  in  the  town 
on  which  the  tax  is  to  be  laid.    If  there  is  a  poll-tax,  a  full 
list  of  the  polls  must  be  made,  and  the  number  multiplied  by 
the  tax  on  each  poll,  and  the  product  must  be  subtracted  from 
the  whole  tax  to  be  raised  by  the  town :   the  remainder  will 
be  the  amount  to  be  raised  on  the  property.     This  being  done, 
the  whole  tax  to  be  raised  must  be  divided  by  the  amount 
of  taxable  property,  and  the  quotient  will  be  the  rate  per 
cent,  of  tax.     Then  this  quotient  must  be  multiplied  by  the 
inventory  of  each  individual,  and  the  product  will  be  the 
tax  on  his  property. 

Examples. 

1.  A  certain  town  is  to  be  taxed  $4280  ;  the  property  on 
which  the  tax  is  to  be  levied  is  valued  at  $1000000.  Now, 
there  are  200  polls,  each  taxed  $1.40.  The  property  of  A 
is  valued  at  $2800,  and  he  pays  4  polls  ; 

B's  at  $2400,  pays  4  polls  ;    E's  at  $7242,       pays  4  polls; 
C's  at  $2530,  pays  2     "         F's  at  $1651,       pays  6     " 
D's  at  $2250,  pays  6     "         G's  at  $1600.80,  pays  4     " 


ASSESSING   TAXES. 


268 


What  will  be  the  tax  on  1  dollar,  and  what  will  be  A's 
tax,  and  also  that  of  each  on  the  list? 

First,       $1.40  x  200  =  $280,  amount  of  poll-tax. 

$4280  —  $280  =  $4000,  amount  to  be  levied  on  property. 

Then,       $4000  -h  $1000000  =  4  mills  on  $1. 

Now,  to  find  the  tax  of  each,  as  A's,  for  example : 


A's  inventory, 

4  polls,  at  $1.40  each, 
A's  whole  tax, 


$2800 
.004 

11.20 
5.60 

$16.80 


In  the  same  manner  the  tax  of  each  person  in  the  town- 
ship may  be  found. 

Having  found  the  per  cent.,  or  the  amount  to  be  raised  on 
each  dollar,  form  a  table  showing  the  amount  which  certain 
sums  would  produce  at  the  same  rate  per  cent.  Thus,  after 
having  found,  as  in  the  last  example,  that  4  mills  are  to  be 
raised  on  every  dollar,  we  can,  by  multiplying  in  succession 
by  the  numbers  1,  2,  3,  4,  5,  6,  7,  8,  &c.,  form  the  following 

Table. 


$   $ 

$      $ 

$      $ 

1  gives  0.004 

20  gives  0.080 

300  gives  1.200 

2   "   0.008 

30   '   0.120 

400   "   1.600 

3   "   0.012 

40   '   0.160 

500   "   2.000 

4   "   0.016 

50   '   0.200 

600   "   2.400 

5   "   0.020 

60   '   0.240 

700   "   2.800 

6   "   0.024 

70   '   0.280 

800   "   3.200 

7   "   0.028 

80   '   0.320 

900   "   3.600 

8   "   0.032 

90   '   0.360 

1000   "   4.000 

9   "   0.036 

100   '   0.400 

2000   "   8.000 

10   "   0.040 

200   '   0.800 

3000   "  12.000 

This  table  shows  the  amount  to  be  raised  on  each  sum  in 
the  column  under  $'s. 


264  PERCENTAGE. 

Examples. 

1.  Find  the  amount  of  B's  tax  from  this  table. 

B's  tax  on  $2000  .  .  is  $8.000 
B's  tax  on  $400  .  .  is  $1.600 
B's  tax  on  4  polls,  at  $1.40,  is  $5.600 

B's  total  tax        .        is       $15.200 

2.  Find  the  amount  of  C's  tax  from  the  table. 

C's  tax  on  $2000  .  .  is  $8.000 

C's  tax  on  500  .  .  is  $2.000 

C's  tax  on  30  .  is  $0.120 

C's  tax  on  2  polls  .  .  is  $2.800 

C's  total  tax         .        is       $12.920 

In  a  similar  manner,  we  might  find  the  taxes   to  be  paid 
by  D,  E,  &c. 

3.  If  the  people  of  a  town  vote  to  tax  themselves  $1500 
to  build  a  public  hall,  and  the  property  of  the  town  is  valued 
at  $300,000,  what  is  D's  tax,  whose  property  is  valued  at 
$2450? 

4.  In  a  school  district  a  school  is  supported  by  a  tax  on 
the  property  of  the  district,  valued  at  $121340.     A  teacher 
is  employed  for  5  months,  at  $40  a  month,  and  contingent 
expenses   are   $42.68 :    what  will  be   a  farmer's   tax,  whose 
property  is  valued  at  $3125? 


CUSTOM-HOUSE  BUSINESS. 

331.  DUTIES  are  sums  of  money  levied  by  government  on 
goods  imported  from  foreign  countries.* 

332.  A  SPECIFIC  DUTY  is  a  certain  sum  on  a  particular 
kind  of  goods  named. 

333.  An  AD  VALOREM  DUTY  is  a  certain  per  cent,  on  the 
cost  of  the  goods  in  the  country  from  which  they  are  imported. 


CUSTOM-HOUSE   BUSINESS.  265 

334.  A  PORT  OF  ENTRY  is  a  port  designated  by  law,  where 
goods  from  a  foreign  country  may  be  landed. 

335.  TONNAGE  is  a  tax  levied  on  vessels,  according  to  their 
size,  for  the  privilege  of  entering  a  port. 

336.  A  CUSTOM-HOUSE  is  an  establishment  created  by  gov- 
3rnment,  at  a  port  of  entry,  for  the  collection  of  duties. 

337.  REVENUE  is  the  income  of  government,  derived  from 
ill  sources.     These  sources  are,  Duties,  Tonnage,  and  Taxes. 

338.  ALLOWANCES  are  deductions  made  from  the  weights 
and  measures  of  goods,  on  account  of  the  bags,  casks,  and 
boxes  which  contain  them. 

339.  GROSS  WEIGHT  is  the  whole  weight  ^of  the  goods,  t(> 
gether  with  that  of  the  casks,  bags,  and  boxes  which  contain 
them. 

340.  DRAFT  is  an  allowance,  from  the  gross  weight,   on 
account  of  waste,  where  there  is  not  actual  tare. 

On  112  Ib it  is  1  Ib. 

From        112  Ib.    to     224  Ib.,       "  2  Ib. 

224  Ib.    to     336  Ib.,       "  3  Ib. 

"  336  Ib.    to    1120  Ib.,       "  4  Ib. 

1120  Ib.    to   2016  Ib.,       "  7  Ib. 

Above    2016  Ib.,     any  weight,  "  9  Ib.; 

consequently,  9  Ib.  is  the  greatest  draft  allowed. 

341.  TARE  is  an  allowance  made  for  the  weight  of  the 
boxes,  barrels,  or  bags  containing  the  commodity,  and  is  of 
three   kinds :    1st,  Legal  tare,  or  such  as   is   established  by 
law ;    2d,  Customary  tare,  or  such  as  is  established  by  the 
custom  among  merchants ;  and  3d,  Actual  tare,  or  such  as  is 
found  by  removing  the  goods  and  actually  weighing  the  boxes 
or  casks  in  which  they  are  contained. 

342.  CUSTOMARY  TARE   on  liquors   in  casks,   is  sometimes 
allowed,  on  the  supposition  that  the  cask  is  not  full,  or  what 
is  called  its  actual  wants;  and  then  an  allowance  of  5  per 
cent.,  for  leakage. 

12 


L>66 


CURRENCY. 


A  tare  of  10  per  cent,  is  allowed  on  porter,  ale,  and  beer, 
in  bottles,  on  account  of  breakage,  and  5  per  cent,  on  all 
other  liquors  in  bottles.  At  the  custorn-house,  bottles  of  the 
common  size  are  estimated  to  contain  2J  gallons  the  dozen. 

NOTE. — For  Tables  of  Tare  and  Duty,  see  Ogden  on  the  Tariff 
of  1842. 

Examples. 

1.  What  will  be  the  duty  on  125  cartons  of  ribbons,  each 
containing  48  pieces,  and  each  piece  weighing  3  oz.  net,  and 
paying  a  duty  of  $2.50  per  pound  ? 

2.  What  will  be  the  duty  on   225   bags   of  coffee,   each 
weighing  gross  1601b.,  invoiced  at  6  cents  per  pound;  2  per 
cent,  being  the  legal  rate  of  tare,  and  20  per  cent,  the  duty  ? 

3.  What  dutyrnust  be  paid  on  275  dozen  bottles  of  claret, 
estimated  to  contain  2|  gallons  per  dozen,  5  per  cent,  being 
allowed  for  breakage,  and  the  duty  being  35  cents  per  gallon  ? 

4.  A   merchant   imports    175   cases    of  indigo,   each   cage 
weighing  196  Ib.  gross ;    15  per  cent,  is  the  customary  rate 
of  tare,  and  the  duty  5  cents  per  pound :  what  duty  must 
he  pay  on  the  whole  ? 


CURRENCY 

343.  CURRENCY  is  what  passes  for  money.     There  are  three 
kinds  of  Currency :   1st,  The  coins  of  the  country ;   2d,  For- 
eign coins,  having  a  fixed  value ;    3d,   Treasury  Notes   and 
Bank  Notes. 

NOTE. — The  foreign  coins  most  in  use  in  this  country,  are  the 
English  shilling,  valued  at  22  cents  2  mills;  the  English  sover- 
eign, valued  at  $4.84;  the  French  franc,  valued  at  18  cents 
0  mills;  and  the  five-franc  piece,  valued  at  $0.93. 

344.  The  Currency  of  the  United  States,  is  the   decimal 
currency  of  dollars,  cents,  and  mills. 

In  some  of  the  States,  the  local  currency  of  pounds,  shil- 
lings, and  pence,  is  yet  preserved. 


ANALYSIS.  267 

The  following  table  shows  the  value  of  a  pound  in  dollars, 
in  each  «urrency  ;  the  number  of  shillings  and  pence  in  a 
dollar,  and  the  value  of  $1  in  pence  : 

In  English  Currency,     £1  =  $4.84,  and  $1  =  4s.  Gd.  -  54d. 
In   N.    E.,    Inda.,    111.,  ) 

Missouri,    Va,,    Ky.,  [  £1  =  $3},  and  $1  =  6s.        =  72d. 

Tenn.,  Miss.,  &  Texas  ) 

-    and  $1=8S-   =%d- 


=  *2t-  •*  *'  =  i*  «•  =  ««. 

In  S.  C.  and  Georgia,      £1  =  $4$,  and  $1  =  4s.  8d.  =  56d. 

1  =  $4-  and  ?1  =  5s-     =  60d- 


The   pound   is   always   reckoned   at   20   shillings,   and   the 
dollar  at  100  cents. 


ANALYSIS. 

345.  An  ANALYSIS  of  a  proposition,  is  an  examination  of 
its  separate  parts,  and  their  connection  with  each  other. 

In  analyzing,  we  reason  from  a  given  number  to  its  unit, 
and  then  from  this  unit  to  the  required  number. 

346.  To  find  the  cost  of  several  things,  when  the  price 
of  1  and  number  are  known. 

1.  If  9  bushels  of  wheat  cost  18  dollars,  what  will  27 
bushels  cost  ? 

ANALYSIS. — One  bushel  costs  |  as  much  as  9  bushels:  £  of  18 
dollars  is  2  dollars.  27"  bushels  will  cost  27  times  as  much  as 
1  bushel ;  $2  x  27  =  $54 ;  therefore,  27  bu.  will  cost  $54. 

OPERATION. 

I  of  $18  =  -V-  =  $2  =  cost  of     1  bushel ; 
$2  x  27  =  $54  =  cost  of  27  bushels. 
'  "NOTE. — The  two  expressions  may  be  combined  in  one ;  thus, 


268 


ANALYSIS. 


2.  If  7  yards  of  cloth  cost  $49,  what  will  16  yards  cost? 

3.  If  12  barrels  of  flour  cost  $72,  what  will  37  bbl.  cost? 

4.  What  will  be  the  cost  of  9J  yards  of  cloth,  if  35  yd. 
cost  $70  ? 

5.  How  many  sheep,   at   4   dollars    a  head,   must  I  give 
for  6  cows,  worth  12  dollars  apiece  ? 

347.  To  find  the  cost  of  articles  in  dollars  and  cents, 
•when  the  price  is  given  in  shillings  and  pence. 

6.  What  will  12  yards  of  cloth^cost,  at  5  shillings  a  yard, 
New  York  currency  ? 

ANALYSIS. — Since  1  yard  costs  5  shillings,  12  yards  will  cost 
12  times  5  shillings,  or  60  shillings :  and  as  8  shillings  make  1 
dollar,  New  York  currency,  there  will  be  as  many  dollars  as  8 
is  contained  times  in  60,  viz.,  7^. 


OPERATION. 


x  5 


=  Y-  =  $f  i  =  $1-50. 


7.  What  will  56  bushels  of  oats  cost,  at  3s.  6d.  a  bushel, 
New  England  currency? 

ANALYSIS.  —  3s.  6d.  =  42d.,  and  72d.  make  1  dollar:  the  cost  in 
pence,  will  be  56  x  42,  and  this  product  divided  by  72,  will  give 
the  answer. 

OPERATION. 


Rule.  —  Multiply  the  commodity  by  the  price,  and  divide 
the  product  by  the  value  of  a  dollar,  expressed  in  the  same 
unit. 

8.  What  will  18  yards  of  satinet  cost,  at  3s.  9d.  a  yard, 
Pennsylvania  currency? 

9.  What  will  7J  Ib.  of  tea  cost,  at  6s.  8d.  a  pound,  New 
England  currency? 

10.  What  will  be  the  cost  of  120  yards  of  cotton  cloth, 
at  Is.  5d.  a  yard,  Georgia  currency? 


ANALYSIS.  269 

11.  What  will  be  the  cost,  in  New  York  currency? 

12.  What  will  be  the  cost,  in  New  England  currency? 

13.  What  will  be  the  cost  of  75  bushels  of  potatoes,  at 
S&.  6d.,  New  York  currency  ? 

14.*What  will  it  cost  to  build  148  feet  of  wall,  at  Is.  8d. 
per  foot,  N.  Y.  currency  ? 

15.  What  will  a  load   of  wheat,  containing  46£  bushels, 
come  to,  at  10s.  8d.  a  bushel,  N.  Y.  currency? 

16.  What  will  7  yards   of  Irish   linen  cost,  at  3s.  4d.  a 
yard,  Penn.  currency? 

17.  How  many  pounds  of  butter,  at  Is.  4d.  a  pound,  must 
be  given  for  12  gallons  of  molasses,  at  2s.  8d.  a  gallon  ? 

18.  What  will  be  the  cost  of  12  cwt.  of  sugar,  at  9d.  per 
lb.,  N.  Y.  currency  ? 

19.  What  will  be  the  cost  of  9  hogsheads  of  molasses,  at 
Is.  3d.  per  quart,  N.  E.  currency? 

20.  How  many  days'  work,  at  7s.  6d.  a  day,  must  be  given 
for  12  bushels  of  apples,  at  3s.  9d.  a  bushel  ? 

348.    Analysis,  when  the  numbers  are  fractional. 

21.  If  3|  pounds  of  tea  cost  $3£,  what  will  9  pounds  cost? 
ANALYSIS.—  3|  lb.  =  \5  lb.,  and  $3£  =  $V°  :  since  Vs  *b-  cost  $  V°i 

1  lb.  will  cost  $V°  -J-  V  =  5°  x  T*T»  and  9  lb-  wil1  cost  9  times  a8 
much,  or     ^  x  T45  x  9. 

OPERATION. 


3 

22.  If  5J  bushels  of  potatoes  cost  $2f,  how  much  will  121 
bushels  cost  ? 

23.  If  1  acre  of  land  costs  |  of  f  of  f  of  $50,  what  will 
3^  acres  cost? 

24.  If  -J  of  f  of  a  gallon  of  wine  costs   f   of  a  dollar, 
what  will  5J  gallons  cost  ? 

,  25.  A  person  purchased  ±  of  a  vessel,  and  divided  it  into 
5  equal  shares,  and  sold  each  of  those  shares  for  $1200: 
what  was  the  value  of  the  whole  vessel  ? 


270 


ANALYSIS. 


26.  What  number  is  that,  f  of  which  is  18? 
ANALYSIS. — Since  -!j-  of  a  number  is  18,  -^  of  it  will  be  -g-  of  18, 

which  is  3:    if  -}   of  a  number  is   3,   the  number  is   equal  to 
7  x  8  =  21. 

27.  32  is  £  of  what  number  ? 

28.  63  is  T72-  of  what  number  ? 

29.  A  traveler,  after  going  196  miles,  found  that  he  had 
performed  T7T  of.  his  journey :  how  long  was  his  journey,  and 
how  much  had  he  yet  to  perform? 

30.  A  father  gave  his  younger  son  $420,  which  was  J  of 
what  he  gave  to  his  elder  son;   and  3  times  the  elder  son's 
portion,  was  \  the  value-  of  the  father's   estate :  what  was 
the  value  of  the  estate  ? 

349.    Proportions  of  Numbers. 

-    31.  If  6  men  can  build  a  boat  in  120  days,  how  long  will 
it  take  24  men  to  build  it  ? 

ANALYSIS.— Six  men,  in  120  days,  will  do  120  x  6  =  720  days' 
work ;  hence,  1  man  would  do  the  work  in  720  days :  24  men 
will  do  the  work  in  ^  of  that  time ;  therefore,  720  -f-  24  =  30  days. 

OPERATION. 

120  x  6  =  720  days  =  time     1  man  can  do  it. 
^°-  —  30  days  —  time  24  men  can  do  it. 

32.  If  6  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  5  men  to  do  it  ? 

33.  If  36  men   can   build   a  house  in  16  days,  how  long 
will  it  take  12  men  to  build  it  ? 

34.  If  3  pipes  can  empty  a  reservoir 'of  water  in  7  days, 
in  what  time  can  8  pipes  empty  it  ? 

35.  If,  by  working   8   hours  a  day,   a   certain   number  of 
men  can  do  a  piece  of  work  in  15  days,  in  what  time  could 
they  do  it,  if  they  work  11  hours  a  day?  * 

36.  A  piece  of  ground  is  32  rods  long,  and  19  rods  wide: 
what  must  be  the  width  of  another  piece  that  shall  have  the 
same  square  measure,  and  whose  length  is  25  rods  ? 

ANALYSIS. — The  square  measure  is  found  by  multiplying  the 
length'  and  breadth;  32  x  19  =  608  sq.  rods:  the  square  measure 


ANALYSIS.  271 

of  the  second  piece  is  the  same,  or  608  sq.  rods,  and  its  length 
'  is  25  rods ;  its  width  must  be  the  quotient  of  608  divided  by  25, 
which  is  24^  rods. 

OPERATION. 

32  x  19  =  608  sq.  rods  =  -» 

area  of  1st  piece.  09  v  i  o 

608  t    Or,    ffJLb!  =  24,'j  rods, 

-r^-  =  24/5  rods  =  width  25 

of  2d  piece. } 

37.  If  a  piece  of  cloth   is  9  feet  long  and  3  feet  wide: 
how  long  must  be  a  piece  of  cloth  that  is  2|  feet  wide,  to 
contain  the  same  number  of  yards  ? 

38.  If  it  take  44  yards  of  carpeting,  that  is  1J  yards  wide, 
to  cover  a  floor :  how  many  yards,  of  |  of  a  yard  wide,  will 
it  take  to  cover  the  same  floor? 

39.  If  a  piece  of  wall-paper,  14  yards  long  and  1  \  feet  wide, 
will  cover  a  certain  piece  of  wall,  how  long  must  another  piece 
be,  that  is  2  feet  wide,  to  cover  the  same  wall  ? 

40.  If  it  takes  5.1  yards  of  cloth,  1.25  yards  wide,  to  make 
a  gentleman's  cloak :  how  much  serge,  £  of  a  yard  wide,  will 
be  required  to  line  it  ? 

41.  If  6  men  can  build  a  wall  80  feet  long,  6  feet  wide, 
and  4  feet  high,  in  15  days,  in  what  time  can  18  men  build 
one  240  feet  long,  8  feet  wide,  and  6  feet  high? 

ANALYSIS. — The  solid  measure  of  the  wall  is  found  by  multi- 
plying the  three  dimensions  together;  80  x  6  x  4  =  1920  cu.  ft 
In  15  days,  1  man  can  do  \  of  1920  =  320  cu.  ft. ;  and  in  1  day, 
he  will  do  TV  of  320  cu.  ft.  =  3T2/.  In  1  day,  18  men  can  do  18 
times  .^j0  =  384  cu.  ft. ;  and  it  will  take  18  men  as  many  days  to 
b\iild  the  wall,  as  384  cu.  ft.  is  contained  times  in  the  solid  measure 
of  the  second  wall:  240  x  8  x  6  =  11520  cu.  ft. .  and  11520—384 
=  30  days. 

OPERATION. 

80  x  6  x  4  =  1920  cu.  ft.  =  solid  measure  of  first  wall. 
£  of  1920  —  320  cu.  ft.  =  work  done  in  15  days  by  1  man. 
*fa  of  320  =  -3r25°-  cu.  ft.  —  work  done  in  1  day  by  1  man. 
-3r25°   X  18  =384  cu.  ft.  =  work  done  in  1  day  by  18  men. 
240  x  8  x  6  =  11520  cu.  ft.  =  solid  measure  of  second  wall. 
=  30  days  =  time  for  18  men  to  do  the  work. 


272 


_ 

" 


42.  If  96  Ibs.  of  bread  be  sufficient  to  serve  5  men  12  days, 
how  many  days  will  57  Ibs.  serve  19  men  ? 

43.  If  a  man  travel  220  miles  in  10  days,   traveling  12 
hours  a  day  :   in  how  many  days  will  he  travel  880  miles, 
traveling  16  hours  a  day  ? 

44.  If  9  men  pay  $135  for  5  weeks'  board,  how  much  must 
8  men  pay  for  4  weeks'  board? 

45.  If  12  men  reap  80  acres  in  6  days,  in  how  many  days 
will  25  men  reap  200  acres  ? 

46.  If  4  men  are  paid  24  dollars  for  3  days'  labor,  how 
many  men  may  be  employed  16  days  for  $96  ? 

47.  A  wall,   to  be  built  to  the  height  of  27   feet,   was 
raised  to  the  height  of  9  feet  by  12  men,  in  6  days  :   how 
many  men  must  be  employed,  to  finish  the  wall  in  4  days,  at 
the  same  rate  of  working  ? 

48.  Two  men  bought  a  horse  for  $150  :  one  paid  $90,  and 
the  other,  $60  ;  they  sold  the  horse,  and  gained  $75  :  what 
did  each  gain  ? 

ANALYSIS.  —  Each  must  have  the  same  part  of  the  gain  that  the 
money  which  he  paid,  is  of  the  whole  money  paid.  One  paid 
$90,  which  is  T9^  =  £  of  $150,  and  he  ought  to  receive  §  of  the 
gain  ;  and  the  other  paid  $60,  which  is  T6/7  =  f  of  $150,  and  he 
ought  to  receive  f  of  the  gain.  |  of  $75  =  $45  =  gain  01  one  ; 
and  |  of  $75  =  $30  =  gain  of  the  other. 

OPERATION. 

$90  =  T9/o  =  I  of  cost  ;   f  of  75  =  $45  =  gain  of  one.    , 
$60  =  T65%  =  f  of  cost  ;   |  of  75  =  $30  =       "        other. 

49.  Three  persons  bought  2  barrels  of  flour  for  15  dollars. 
The  first  one  ate  from  them  2  months,  the  second  3  months, 
and  the  third  7  months  :    how  much  should  each  pay  ? 

50.  If  two  persons  engage  in  a  business,  where  one  ad- 


ANALYSIS.  273 

vances  $875  and  the  other  $625,  and  they  gain  $300,  what 
is  each  one's  share  ? 

51.  A,  B,  arid  C  sent  a  drove  of  hogs  to  market,  of  which 
A  owned  105,  B  75,  and  C  120  ;  on  the  way,  60  died:   how 
many  must  each  lose  ? 

52.  A  man  who  has   only  $50,  owes  $75  to  A,  $150  to 
B,  and  $100  to  C  :    how  much  ought  he  to  pay  to  each  ? 

53.  A  can  do  a  piece  of  work  in  4  days,  and  B  can  do 
the   same  in   6   days  :    in  what  time  can   they  both  do   the 
work,  if  they  labor  together  ? 

ANALYSIS.  —  Since  A  can  do  the  work  in  4  days,  in  1  day  he 
can  do  |  of  the  work,  and  B  can  do  £  of  the  work  in  1  day: 
both  can,  in  1  day,  do  the  sum  of  |  and  |:  |  4-  £  =  J°  —  ^  : 
since  in  1  day  they  can  do  ^  of  the  work,  it  will  require,  for 
the  whole  work,  as  many  days  as  ^  is  contained  times  in  1. 
1  ~  &  =  I  x  V  =  V  =  2f  days. 

OPERATION. 

1  =  what  A  can  do  in  1  day  ; 
i  =  what  B 
i  +  i  =  ^  =  j\  =  what  both  can  do  in  1  day  ; 

!    .    5    ._  i  v  12  _  i  2  _  02  fi    ,„  j  required  for  A  and  B  to 
1  -  T*  -    X  -5-  -  -  -5-  -  *s 


54.  A  can  build  a  shed  in  6  days,  and  B  can  build  it  in 
5  days  :   in  what  time  can  they,  by  working  together,  build 
the  shed  ? 

55.  A  father  earns,  in  9  days,  $18,  and  his  son  earns  the 
same  amount  in  15  days  :  in  what  time  could  they,  together, 
earn  the  amount  ? 

56.  A  laborer  can  dig  a  trench  in  25  days,  but  with  the 
assistance  of  a  second  laborer,  he  digs  it  in  16  days  :  in  what 
time  would  the  second  laborer,  alone,  have  dug  it  ? 

57.  A  can  build  a  wall  in  16  days,  and   B  can  do  it  in 
*21   days  ;   they  both  worked  on   the  wall  ;   after  working  5 
days,  B  left  it  :   in  what  time  could  A  finish  the  work  ? 

58.  A  can  build  a  wall  in  18  days,  and  B  can  do  it  in 
24  days  ;   A  worked  alone  for  6  days,  and  was  then  assisted 
by  B  :   in  what  tune  was  the  work  finished  ? 

12* 


274  ANALYSIS. 

59.  If  a  barrel  of  flour  would  last  a  family  6  weeks,  and 
if  it  would  last  a  second  family  8  weeks :  how  long  would  \ 
of  the  barrel  last  both  families  ? 

60.  Divide  $500  between  3  persons,  giving  to  one  $J,  as 
often  as  to  the  second  $J,  and  to  the  third  $£. 

61.  Divide   $176.40   among   3   persons,   so   that   the  first 
shall  have  twice  as  much  as  the  second,  and  the  third  three 
times  as  much  as  the  first :  what  is  each  one's  share  ? 

62.  A  person  bought  3  lots  of  ground  for  $6000 ;  he  paid 
$150  more  for  the  second  than  for  the  first,  and  $350  more 
for  the  third  than  for  the  second  :  what  was  the  cost  of  each  ? 

63.  Three  men  hire  a  pasture,  for  which  they  pay  66  dol- 
lars.    The  first  puts  in  2  horses  for  3  weeks  ;  the  second,  6 
horses  for  2J  weeks;  the  third,  9  horses  for  1^  weeks:  how 
much  ought  each  to  pay? 

ANALYSIS. — The  pasturage  of  2  horses  for  3  weeks,  would  "be 
the  same  as  the  pasturage  of  1  horse  2  times  3  weeks,  or  6 
weeks ;  that  of  six  horses,  2|  weeks,  the  same  as  for  1  horse  six 
times  2-*-  weeks,  or  15  weeks;  and  that  of  9  horses  \\  weeks,  the 
same  as  1  horse  for  9  times  1£  weeks,  or  12  weeks.  The  three 
persons  had  an  equivalent  for  the  pasturage  of  1  horse  for  6  +  15 
+  12  —  33  weeks ;  therefore,  the  first  must  pay  J5 ;  the  second,  if ; 
and  the  third,  if  of  68  dollars. 

OPERATION. 

3  X  2  =  6  ;  then,  $66  x  A  =  $12.  1st. 
21  x  6  =  15  ;  "  866  x  J*  =  $30.  2d. 
1 J  X  9  =  12  ;  "  $66  X  l|  =  $24.  3d. 

64.  Two  persons,  A  and  B,  enter  into  partnership,  and  gain 
$175.     A  puts  in  75  dollars  for  4  months,  and  B  puts  in  100 
dollars  for  6  months  :  what  is  each  one's  share  of  the  gain  ? 

65.  Three  men  engage  to  build  a  house  for  580  dollars. 
The  first  one  employed  4  hands  ;  the  second,  5  hands  ;   and 
the  third,  7  hands.     The  first  man's  hands  worked  3  times  as 
many  days  as  the  third,  and  the  second  man's  hands  twice  as 
many  days  as  the  third  man's  hands :   how  much  must  each 
receive  ? 


ALLIGATION.  275 


ALLIGATION. 

350.  ALLIGATION  is   the   process   of  mixing   substances   in 
such  a  manner  that  the  value  of  the  compound  may  be  equal 
to  the  sum  of  the  values  of  the  several  ingredients. 

ALLIGATION    MEDIAL. 

351.  ALLIGATION  MEDIAL  is  the  process  of  finding  the  mean 
price  of  a  mixture,  when  the  quantity  of  each  simple  and  its 
price,  are  known. 

I.  A  merchant  mixes  8  Ib.  of  tea,  worth  75  cents  a  pound, 
with  16  Ib.,  worth  $1.02  a  pound:  what  is  the  price  of  the 
mixture  per  pound? 

ANALYSIS— The  quantity,  8  Ib.  of  OPERATION. 

tea,  at  75  cents  a  pound,  costs  $6;  8  Ib.  at  75  cts.  =  $  6.00 

and  16  Ib.,  at  $1.02,  costs  $16.32:  16  Ib.  at    $1.02  =  $16.32 

hence,  the  mixture,  =  24  Ib.,  costs  9 .                          91  x  99  q9 
$22.32  ;    and  the  price  of  1  Ib.  of 

the  mixture  is  found  by  dividing  $0.93 

this  cost  by  24 :     Hence,  to  find  the  price  of  the  mixture, 

Rule. — I.   Find  the  cost  of  the  entire  mixture : 

II.  Divide  the  entire  cost  of  the  mixture  by  the  sum  of 
the  simples,  and  the  quotient  will  be  the  price  of  the  mixture. 

Examples. 

1.  A  farmer  mixes  30  bushels  of  wheat,  at  5s.  per  bushel, 
and  72  bushels  of  rye,  at  3s.,  with  60  bushels  of  barley,  at  2s. : 
what  is  the  price  of  1  bushel  of  the  mixture  ? 

2.  A  wine-merchant  mixes  15  gallons  of  wine,  at  $1  per 
gallon,  with  25  gallons  of  brandy,  worth  75  cents  per  gallon : 
what  should  be  the  price  of  a  gallon  of  the  compound  ? 

3.  A  grocer  mixes  40  gallons  of  whisky,  worth  31  cents 
per  gallon,   with   3   gallons   of  water,   which  costs  nothing : 
what  should  be  the  price  of  a  gallon  of  the  mixture  ? 

4.  A  goldsmith  melts  together  2  Ib.  of  gold,  of  22  carats 


276 


ALLIGATION. 


fine,  6  oz.,  of  20  carats  fine,  and  6  oz.,  of  16  carats  fine: 
what  is  the  fineness  of  the  mixture  ? 

5.  On  a  certain  day,  the  mercury  in  the  thermometer  was 
observed  to  average  the  following  heights :  from  6  in  the 
morning  to  9,  64°;  from  9  to  12,  74°;  from  12  to  3,  84°; 
and  from  3  to  6,  70°:  what  was  the  mean  temperature  of 
the  day? 

ALLIGATION  ALTERNATE. 

352.  ALLIGATION  ALTERNATE  is  the  process  of  finding  what 
proportions  must  be  taken  of  each  of  several  simples,  whose 
prices  are  known,  to  form  a  compound  of  a  given  price.  It 
is  the  opposite  of  Alligation  Medial,  and  may  be  proved  by  it. 

Alligation  Alternate  is  founded  on  an  equality  of  Gain  and 
Loss.  In  selling  a  mixture  at  a  mean  price,  there  is  a  gain 
on  each  simple  below  that  price,  and  a  loss  on  each  simple 
above  the  average  price.  The  gain  must  be  just  equal  to  the 
loss,  otherwise  the  value  of  the  compound  would  not  be  a 
mean  value. 

CASE    i. 

353.    To  find  the  proportional  parts. 

1.  A  farmer  would  mix  oats  at  3s.  a  bushel,  rye  at  6s., 
and  wheat  at  9s.  a  bushel,  so  that  the  mixture  shalt  be 
worth  5  shillings  a  bushel :  what  proportion  must  be  taken 
of  each  sort  ? 

ANALYSIS. — Having  written  the  simples  in  the  order  of  their 
values,  place  the  mean  price  at  the  left,  and  then  rule  5  columns. 
The  column  U  expresses  that  1  unit  is  first  taken  of  e$,ch  simple ; 
the  column  G  expresses  the  gain  on  1  unit;  the  column  L,  the 
loss  on  1  unit;  and  the  difference  between  the  sums  of  the  col- 
umns, shows  that  on  1  unit  of  each  simple,  there  is  a  loss  of  3. 

Since  the  column  of  gains  must  balance  the  column  of  losses, 
we  must  take  so  many  additional  units  as  will  give  a  gain  of  3s. 
Since  1  unit  gives  a  gain  of  2s.,  |  of  a  unit  will  give  a  gain  of 
Is.,  and  f  units  will  give  a  gain  of  3s. :  hence,  1  +  f  =  f  will 
give  a  gain  of  5s.,  which  will  balance  the  losses.  Write  f  in  the 
column  Bal.,  and  then  multiply  each  proportional  number  by  2 
(the  denominator  of  the  fraction),  which  will  give  the  proportional 
parts,  5,  2,  and  2,  in  integral  numbers. 


ALLIGATION   ALTERNATE. 


277 


OPEIIATION. 


Oats,        3 
Rye,          6 
Wheat,     9 

G. 

L. 

u. 

Bal. 

P.P. 

2 

1 

4 

1 
1 

1 

5 
T 
1 
1 

5 
2 

2 

2        5 ;       5-2  =  3. 

Rule. — I.  Write  the  prices  of  the  simples  in  a  column, 
beginning  with  the  lowest,  and  the  mean  price  at  the  left. 

II.  Opposite  each  simple,  write  its  gain  or  loss  on  1,  and 
write  the  1  in  the  column  of  units  : 

III.  Find  the  difference  between  the  gains  and  losses,  and 
divide  it  by  the  first  gain  or  loss  in  the  least  column,  and 
add  the  quotient  to  the  corresponding  number  in  the  column 
U,  and  then  write  the  sum  and  the  other  numbers  of  that 
column,  in  the  column  Bal.: 

IY.  When  the  quotient  is  fractional,  multiply  each  num.- 
ber  by  the  denominator  of  the  fraction,  and  the  several 
products  will  be  the  proportional  parts  in  integral  numbers, 
which  write  in  column  P.  P.  of  Proportional  Parts. 

NOTE. — The  answers  to  the  last,  and  to  all  similar  questions, 
will  be  infinite  in  number.  For,  if  the  proportional  numbers  in 
column  P.  P.  be  multiplied  by  any  number,  integral  or  .fractional, 
the  products  will  denote  proportional  parts  of  the  simples. 

Examples. 

1.  What  proportions  of  tea,  at  24  cents,  30  cents,  33  cents, 
and  36  cents  a  pound,  must  be  mixed  together  so  that  the 
mixture  shall  be  worth  32  cents  a  pound? 

2.  What  proportions  of  coffee,  at  16  cts.,  20  cts.,  and  28 
cts.  per  pound,  must  be  mixed  together  so  that  the  compound 
shall  be  worth  24  cts.  per  pound  ? 

3.  A  goldsmith  has  gold  of  16,  of  18,  of  23,  and  of  24 
carats  fine :    what  part  must  be  taken  of  each,  so  that  the 
mixture  shall  be  21  carats  fine? 


278  ALLIGATION. 

4.  What  portion   of  brandy,  at  14s.  per  gallon;    of  old 
Madeira,   at  24s.  per  gallon;   of  new  Madeira,   at  21s.  per 
gallon  ;   and  of  brandy,  at  10s.  per  gallon,  must  be  mixed  to- 
gether so  that  the  mixture  shall  be  worth  18s.  per  gallon? 

5.  How  many  gallons  of  water  must  1?e  mixed  with  wine 
at  83  per  gallon,  and  wine  at  $5  per  gallon,  that  the  mix- 
ture may  be  sold  at  $2  per  gallon? 

CASE    II. 
354.    "When  the  quantity  of  one  simple  is  given. 

I.  How  much  wheat,  at  9s.  a  bushel,  must  be  mixed  with 
20  bushels  of  oats,  worth  3  shillings  a  bushel,  that  the  mix- 
ture may  be  worth  5  shillings  a  bushel? 

ANALYSIS. — First,  find  the  proportional  numbers,  as  in  the  last 
case :  they  are,  2  of  oats  and  1  of  wheat.  Hence,  for  every  2 
bushels  of  oats,  we  take  1  of  wheat ;  and  since  we  take  20  bushels 
of  oats,  we  take  as  many  of  wheat  as  is  contained  times  in  20, 
which  is  10. 

Rule. — I.  Find  the  proportional  numbers,  and  divide  the 
given  quantity  by  its  proportional  number: 

II.  Multiply  each  remaining  proportional  number  by  this 
ratio,  and  the  products  will  denote  the  quantities  to  be  taken 
of  each. 

Examples. 

1.  How  much  wine,  at  5s.,  at  5s.  6d.,  and  6s.  per  gallon, 
must  be  mixed  with  4  gallons,  at  4s.  per  gallon,  so  that  the 
mixture  shall  be  worth  5s.  4d.  per  gallon? 

2.  A  farmer  would  mix  14  bushels  of  wheat,  at  $1.20  per 
bushel,  with  rye  at  72  cts.,  barley  at  48  cts.,  and  oats  at  36 
cts. :  how  much  must  be  taken  of  each  sort  to  make  the  mix- 
ture worth  64  cts.  per  bushel  ? 

3.  There  is  a  mixture  made  of  wheat  at  4s.  per  bushel, 
rye  at  3s.,  barley  at  2s.,  with  12  bushels  of  oats  at  18d.  per 
bushel :  how  much  is  taken  of  each  sort  when  the  mixture  is 
worth  3s.  6d.  ? 


ALLIGATION  ALTERNATE.  279 

4.  A  distiller  would  mix  40  gallons  of  French  brandy,  at 
12s.  per  gallon,  with  English  at  7s.,  and  spirits  at  4s.  per  gal. : 
what  quantity  must  be  taken  of  each  sort,  that  the  mixture 
may  be  afforded  at  8s.  per  gallon? 

CASE    in. 
355.   When  the  quantity  of  the  mixture  is  given. 

1.  A  merchant  would  make  up  a  cask  of  wine,  containing 
50  gallons,  with  wine  worth  16s.,  18s.,  and  22s.  a  gallon,  hi 
such  a  way  that  the  mixture  may  be  worth   20s.  a  gallon : 
how  much  must  he  take  of  each  sort  ? 

ANALYSIS. — Find  the  proportional  parts:  they  arc,  1,1,  and  3. 
Now,  these  numbers  must  be  taken  as  many  times  as  their  sura,  5, 
is  contained  times  in  50,  which  is  10:  hence,  there  are  10  gallons 
of  the  first,  10  of  the  second,  and  30  of  the  third :  Hence, 

Rule. — I.  Find  the  proportional  parts : 

II.  Divide  the  quantity  of  the  mixture  by  the  sum  of  the 
proportional  parts  : 

III.  Multiply  this    ratio   by   the  parts    separately,    and 
each  product  will  denote  the  quantity  of  the  corresponding 
simple. 

Examples. 

1:  A  grocer  has  four  sorts  of  sugar,  worth  12d.,  10d.,  6d., 
and  4d.  per  pound ;  he  would  make  a  mixture  of  144  pounds, 
worth  8d.  per  pound :  what  quantity  must  be  taken  of  each 
sort  ? 

2.  A  grocer,  having  four  sorts  of  tea,  worth  5s.,  6s.,  8s., 
and  9s.  per  pound,  wishes  a  mixture  of  87  pounds,  worth  7s. 
per  pound :  how  much  must  he  take  of  each  sort  ? 

3.  A  silversmith  has  four  sorts  of  gold,  viz.,  of  24  carats 
fine,  "of  22  carats  fine,  of  20  carats  fine,  and  of  15  carats 
fine  ;  he  would  make  a  mixture  of  42  oz.,  of  17  carats  fine: 
how  much  must  be  taken  of  each  sort  ? 

PROOF. — All  the  examples  of  Alligation  Medial  may  be 
proved  by  Alligation  Alternate. 


280 


INVOLUTION. 


INVOLUTION. 

356.  The  POWER  of  a  number,  is  the  product  which  arises 
from  multiplying  the  number  successively  by  itself.     The  num- 
ber, so  multiplied,  is  called  the  root  of  the  power. 

The  first  power  is  the  number  itself,  or  the  root : 
The  second  power  is  the  product  of  the  root  by  itself : 
The  third  power  is  the  product  when  the  root  is  taken  3 
times  as  a  factor : 

The  fourth  power,  when  it  is  taken  4  times : 
The  fifth  power,  when  it  is  taken  5  times,  &c. 

357.  The  EXPONENT  of  a  power,  is  the  number  denoting 
how  many  tunes  the  root  is  taken  as  a  factor.     It  is  written 
a  little  at  the  right,  and  over  the  root :   thus,  if  the  equal 
factor  or  root  is  4,  ^       4>  the  lst  power  of  4. 

42=  4x4=     16,  the  2d  power  of  4. 

43=  4  x  4  x  4  —     64,  the  3d   power  of  4. 

44=  4x4x4x4=  256,  the  4th  power  of  4. 

45=  4x4x4x4x4  =1024,  the  5th  power  of  4. 

358.  INVOLUTION  is  the  process  of  finding  the  powers  of 
numbers. 

There  are  three  things  connected  with  every  power :  1st,  The 
root ;  2d,  The  exponent ;  and  3d,  The  power  or  result  of  the  mul- 
tiplication. 

In  finding  a  power,  the  root  is  always  the  first  power ;  hence, 
the  number  of  multiplications  is  1  less  than  the  exponent. 

Rule. — Multiply  the  number  by  itself  as  many  times, 
less  1,  as  there  are  units  in  the  exponent,  and  the  last 
product  will  be  the  power. 


1.  Square  of  J. 

2.  Cube  of  |. 

3.  Cube  of  12. 

4.  3d   power  of  125. 

5.  4th  power  of  9. 

6.  5th  power  of  16 


Examples. 

7.  2d  power  of  225. 

8.  Cube  of  321. 

9.  4th  power  of  215. 
10.     Square  of  36049. 

Cube  of  .25. 


11. 

12.     4th  power  of  8.638. 


EVOLUTION.  281 


EVOLUTION. 

359.  EVOLUTION  is  the  process  of  finding  the  equal  factor 
when  we  know  the  power. 

The  square  root  of  a  number  is  the  factor  which,  multiplied 
by  itself  once,  will  produce  the  number :  thus,  6  is  the  square 
root  of  36,  because  6  X  6  =  36. 

The  cube  root  of  a  number  is  the  factor  which,  multiplied 
by  itself  twice,  will  produce  the  number :  thus,  3  is  the  cube 
root  of  27,  because  3  x  3  x  3  =  27. 

The  sign  y',  is  called  the  radical  sign.  When  placed  be- 
fore a  number,  it  denotes  that  its  square  root  is  to  be  ex- 
tracted :  Thus,  V36  =  6. 

We  denote  the  cube  root  by  the  same  sign,  by  writing  3 
over  it :  thus,  V27,  denotes  the  cube  root  of  27,  which  is 
equal  to  3.  The  small  figure,  3,  placed  over  the  radical,  is 
the  index  of  the  root. 

EXTRACTION  OF  THE  SQUARE  ROOT. 

360.  The  SQUARE  ROOT  of  a  number,  is  a  factor  which, 
multiplied  by  itself  once,  will  produce  the  number.     To  ex- 
tract the  square  root,  is  to  find  this  factor.     The  first  ten 
numbers,  and  their  squares,  are, 

1,       2,       3,       4,       5,       6,       7,       8,       9,       10. 
1,       4,       9,      16,     25,     36,     49,     64,     81,     100. 

The  numbers  in  the  first  line  are  the  square  roots  of  the 
numbers  in  the  second :  hence,  the  square  root  of  any  number 
expressed  by  two  figures,  will  be  expressed  by  one  figure. 

361.  A  PERFECT  SQUARE  is  a  number  which  has  two  exact 
factors.     Thus,  1,  4,  9,  16,  25,  36,  &c.,  are  perfect  squares. 

NOTE. — The  square  root  of  a  number  less  than  100  will  be  less 
than  10,  while  the  square  root  of  a  number  greater  than  100  will 
be  greater  than  10. 


282 


EVOLUTION. 


352.   What  is  the  square  of  36  =  3  tens  +  6  units? 

3  +  6 
3  +  6 

3  X  6  +  62 


ANALYSIS.— 36  =  3  tens  +  6  units,  is 
first  to  be  taken  6  units'  times,  giving 
6*+  3  x  6:  then,  taking  it  3  tens'  times, 


we   have   3  x  6  +  32,    and  the    sum  is, 
33  +  2  (3  x  G)  +  62:  that  is, 


32  +  3  X  6 


2(3x6)  +  6 


The  square  of  a  number  is  equal  to  the  square  of  the  tens 
phis  twice  the  product  of  the  tens  by  the  units,  plus  th^ 
square  of  the  units. 

The  same  may  be  shown  by  the  figure : 

Let  the  line  AB  repre-  30  i 

sent  the  3  tens  or  30,  and 
BO  the  six  units. 

Let  AD  be  a  square  on 
AC,  arid  AE  a  square  on 
the  ten's  line  AB. 

Then  E  D  will  be  a 
square  on  the  unit  line  6, 
and  the  rectangle,  E  F,  will 
be  the  product  of  H  E,  which 
is  equal  to  the  ten's  line, 
by  IE,  which  is  equal  to 
the  unit  line.  Also,  the 
rectangle  BK  will  be  the 
product  of  E  B,  which  is 

equal  to  the  ten's  line,  by  the  unit  line,  BO.  But  the  whole 
square  on  A  C  is  made  up  of  the  square  A  E,  the  two  rectangles, 
FE  and  EO,  and  the  square  ED. 

1.  Let  it  now  be  required  to  extract  the  square  root  of 
1296. 

ANALYSIS. — Since  the  number  contains  more  than  two  places 
of  figures,  its  root  will  contain  tens  and  units.  But  as  the  square 
of  one  ten  is  one  hundred,  it  follows  that  the  square  of  the  tens  of 
the  required  root  must  be  found  in  the  two  figures  on  the  left  of 
96  Hence,  we  point  off  the  number  into  periods  of  two  figures 
each,  giving  12  tens,  and  96  units. 


30 
6 
180 

0 
6 
36 

.    30                 E 
900+180+180+36=1296. 

30 
30 

900 

SQUAKK    ROOT.  283 

Tho  greatest  perfect  square  in  12  tens,  is  9  OPERATION. 

tens,  the  root  of  which  is  3  tens,  or  80.     We  ^2  96  (  36 

square  3  tens,  which   gives  9  hundred,  place  9 

9  under  the  hundreds'  place,  and  subtract ; 
this  takes  away  the  square  of  the  tens,  and 
leaves  396,  which  is  twice  the  product  of  the 
tens  by  the  units  plus  the  square  of  the  units. 

If  now,  we  double  the  divisor,  and  then  divide  this  remainder, 
exclusive  of  the  right-hand  figure  (since  that  figure  can  not  enter 
into  the  product  of  the  tens  by  the  units),  by  it,  the  quotient  will 
be  the  units .  figure'  of  the  root.  If  we  annex  this  figure  to  the 
augmented  divisor,  and  then  multiply  the  whole  divisor,  thus  in- 
creased, by  it,  the  product  will  be  twice  the  tens  by  the  units  plus 
the  square  of  the  units;  and  hence,  we  have  found  both  figures 
of  the  root. 

Hence,  for  the  extraction  of  the  square  root,  we  have  the  following 

Rule. — I.  Separate  the  given  number  into  periods  of  two 
figures  each,  by  setting  a-  dot  over  the  place  of  units,  a 
second  over  the  place  of  hundreds,  and  on  each  alternate 
figure  at  the  left: 

II.  Note  the  greatest  square  contained  in  the  period  on 
the  left,  and  place  its  root  on  the  right,  after  the  manner 
of  a  quotient  in  division.     Subtract  the  square  of  this  root 
from  the  first  period,  and  to  the  remainder  bring  down  the 
second  period  for  a  dividend  : 

III.  Double  the  root  thus  found  for  a  trial  divisor,  and 
place  it  on  the  left  of  the  dividend.     Find  how  many  times 
the  trial  divisor  is  contained  in  the  dividend,  exclusive  of 
the  right-hand  figure,  and  place  the  quotient  in  the  root, 
and  also  annex  it  to  the  divisor: 

IV.  Multiply  the  divisor  thus  increased,  by  the  last  figure 
of  the  root ;  subtract  the  product  from  the  dividend,  and  to 
the  remainder  bring  down  the  next  period  for  a  new  divi- 
dend : 

V.  Double  the  whole  root  thus  found,  for  a  new  trial 
divisor,  and  continue  the  operation  as  before,  until  all  the 
periods  are  brought  down. 


284  EVOLUTION. 

NOTEH. — 1.   The  left-hand  period  may  contain  but  one  figure  ; 
each  of  the  others  will  contain  two. 

2.  If  any  trial  divisor  is  greater  than  its  dividend,  the  corre- 
sponding quotient  figure  will  be  a  cipher. 

3.  If  the  product  of  the  divisor  by  any  figure  of  the  root  ex- 
ceeds the  corresponding  dividend,  the  quotient  figure  is  too  large, 
and  must  be  diminished. 

4.  There   will  be  as  many  figures  in  the  root  as  there   are 
periods  in  the  given  number. 

5.  If  the  given  number  is  not  a  perfect  square,  there  will  be 
a  remainder  after  all  the  periods  are  brought  down.     In  this  case, 
periods  of  ciphers  may  be  annexed,  forming  new  periods,  each  of 
which  will  give  one  decimal  place  in  the  root. 

Examples. 

1.  What  is  the  square  root  of  263169? 

ANALYSIS. — "We  first  place  a  dot  over  OPERATION. 

the  9,  making  the  right-hand  period  69.  26  3l   69  (  513 

We  then  put  a  dot  over  the  1,  and  also 
over  the  6,  making  three  periods. 


The  greatest  perfect  square  in  26,  is       101  )  18. 

25,  the  root  of  which  is  5.     Placing  5 

in  the  root,  subtracting  its  square  from       1023)  3069 

26,  and  bringing  down  the  next  period,  3069 
31,  we  have  131  for  a  dividend,  and. by 

doubling  the  root,  we  have  10  for  a  trial  divisor.  Now,  10  is 
contained  in  13,  1  time.  Place  1  both  in  the  root  and  in  the 
divisor:  then  multiply  101  by  1;  subtract  the  product,  and  bring 
down  the  next  period. 

We  must  now  double  the  whole  root,  51,  for  a  new  trial  divi- 
sor; then  dividing,  we  obtain  3,  the  third  figure  of  the  root. 

2.  What  is  the  square  root  of  36729? 

3.  What  is  the  square  root  of  2125? 

4.  What  is  the  square  root  of  26883881? 

5.  What  is  the  square  root  of  426409  ? 

6.  What  is  the  square  root  of  2976412? 

7.  What  is  the  square  root  of  91874261  ? 


SQUARE    ROOT. 


285 


363.    To  extract  the  square  root  of  a  fraction. 
1.  What  is  the  square  root  of  .5  ? 


NOTE.—  We  first  annex  one 
cipher,  to  make  even  decimal 
places.  We  then  extract  the 
root  of  the  first  period  :  to  the 
remainder  we  annex  two  ci- 
phers,  forming  a  new  period, 
and  so  on. 

2.  What  is  the  square  root  of 

NOTE.  —  The  square  root  of  a 
fraction  is  equal  to  the  square 
root  of  the  numerator  divided 
by  the  square  root  of  the  de- 
nominator. 


OPERATION. 

.50  (.707  + 


140  \ 


000 


151,  Rem. 


OPERATION. 


/-  /- 

\l  _  —      _  —  -  _ 
*  - 


OPERATION. 

^  —   75  • 

Vf  —  V  .75  =  .8660  +  . 


3.  What  is  the  square  root  of 

NOTE.  —  When  the  terms  are 
not  perfect  squares,  reduce  the  ' 
common  fraction  to  a  decimal 
fraction,   and  then  extract  the 
square  root  of  the  decimal. 

Rule.  —  I.  If  the  fraction  is  a  decimal,  point  off  the 
periods  from  the  decimal  point  to  the  right,  annexing  ciphers 
if  necessary,  so  that  each  period  shall  contain  two  places, 
and  then  extract  the  root  as  in  integral  numbers: 

II.  If  the  fraction  is  a  common  fraction,  and  its  terms 
perfect  squares,  extract  the  square  root  of  the  numerator 
and  denominator  separately  ;  if  they  are  not  perfect  squares, 
reduce  the  fraction  to  a  decimal,  and  then  extract  the  square 
root  of  the  decimal. 

Examples. 

What  are  the  square  roots  of  the  following  numbers: 


1.  Of  3? 

2.  Of   11? 

3.  Of   1069? 


Of  2268741? 
Of  7596796? 
Of  36372961 ? 


286 


EVOLUTION. 


7. 

Of 

22071204? 

14. 

Of 

4.426816  ? 

8. 

Of 

3271.4207? 

15. 

Of 

8f  ? 

9. 

Of 

4795.25731  ? 

16.* 

Of 

9f? 

10. 

Of 

4.372594  ? 

17: 

Of 

64    ? 
125* 

11. 

Of 

.0025  ? 

18. 

Of 

125   ? 

7  29   * 

12. 

Of 

.00032754  ? 

19. 

Of 

2304  ? 
5184  ' 

13. 

Of 

.00103041? 

20. 

Of 

2704   ? 
4225    * 

Applications  in  Square  Root. 

364.   A  TRIANGLE  is  a  plain  figure,  which  has  three  sides 
and  three  angles. 

If  a  straight  line  meets  another  straight 
line,  making  the  adjacent  angles  equal,  each  is 
called,  a  right  angle ;  and  the  lines  are  said  to 
be  perpendicular  to  each  other. 


365.  A  RIGHT-ANGLED  TRIANGLE  is  one 
which  has  one  right  angle.  In  the  right-angled 
triangle  ABC,  the  side  AC,  opposite  the  right 
angle  B,  is  called,  the  hypothenuse;  the  side 
AC,  the  base;  and  the  side  BC,  the  perpen- 
dicular. 


366.  We  have  seen  (Art.  196),  that  the  area  of  a  square 
is  equal  to  the  product  of  two   of  its  equal  sides,  which  is 
the  square  of  one  side.     Hence,  the  square  root  of  the  area 
of  a  square,  will  be  the  side  itself. 

Thus,  if  the  area  of  the  square  in  the  figure 
is  25,  the  square  root  of  25,  which  is  5,  will 
denote  the  side. 

367.  In  a  right-angled  triangle,  the   square   described  on 
the  hypothenuse  is  equal  to  the  sum  of  the  squares  described 
on  the  other  two  sides. 


SQUARE   ROOT. 


287 


Thus,  if  AOB  be  a  right- 
angled  triangle,  right-angled  at 
C,  then  will  the  large  square,  D, 
described  on  the  hypothenuse 
AB,  be  equal  to  the  sum  of  the 
squares  F  and  E,  described  on 
the  sides  AC  and  CB.  This  is 
called,  the  carpenter's  theorem. 
By  counting  the  small  squares 
in  the  large  square,  D,  you  will 
find  their  number  equal  to  that 
contained  in  the  small  squares 
F  and  E.  In  this  triangle,  the 
hypothenuse  AB  =  5,  AC  =  4, 

and  CB  =  3.  Any  numbers  having  the  same  ratio  as  5,  4,  and 
3,  such  as  10,  8,  and  6 ;  20,  16,  and  12,  &c.,  will  represent  the 
sides  of  a  right-angled  triangle. 

1.  Wishing  to  know  the  distance  from  A  to  the  top  of  a 
tower,  I  measured  the  height  of  the  tower,  and  found  it  to 
be  40  feet ;  also  the  distance  from  A  to  B,  and  „ 
found  it  30  feet:  what  was  the  distance  from 

A  to  C? 

AB  =  30;     AB2  =  30'=    900 
BC  =  40  ;      BC>  -  40'  =  1600 
AC2  =  AB5  +  CB1  =  2500 

AC  =  V2500  =  50  feet.  4~~          "  B 

Hence,  when  the  base  and  perpendicular  are  known,  and 
the  hypothenuse  is  required, 

Square  the  base  and  square  the  perpendicular,  add  the 
results,  and  then  extract  the  square  root  of  their  sum. 

2.  What  is  the  length  of  a  rafter  that  will  reach  from  the 
eaves  to  the  ridge-pole  of  a  house,  when  the  height  of  the 
roof  is  15  feet,  and  the  width  of  the  building,  40  feet  ? 

368.    To  find  one  side,  when  we  know  the  hypothenuse 
and  other  side. 

1.  The  length  of  a  ladder  which  will  reach  from  the  mid 


288  EVOLUTION. 

<|dle  of  a  street,  80  feet  wide,  to  the  eaves  of  a  house,  is  50 
feet :  what  is  the  height  of  the  house  ? 

ANALYSIS. — Since  the  square  of  the  length  of  the  ladder  is 
equal  to  the  sum  of  the  squares  of  half  the  street  and  the  height 
of  the  house,  the  square  of  the  length  of  the  ladder  diminished 
by  the  square  of  half  the  street,  will  be  equal  to  the  square  of 
the  height  of  the  house :  Hence, 

Square  the  hypothenuse  and  the  known  side,  and  take  the 
difference ;  the  square  root  of  the  difference  will  be  the  other 


Examples. 

1.  If  an  acre  of  land  be  laid  out  in  a  square  form,  what 
will  be  the  length  of  each  side  in  rods  ? 

2.  What  will  be  the  length  of  the   side   of  a  square,  in 
rods,  that  shall  contain  100  acres  ? 

3.  A  general  has  an  army  of  7225  men:  how  many  must 
be  put  in  each  line,  hi  order  to  place  them  in  a  square  form  ? 

4.  Two  persons   start  from  the  same  point ;   one  travels 
due  east  50  miles,  the  other  due  south  84  miles :   how  far 
are  they  apart  ? 

5.  What  is  the  length,  in  rods,  of  one  side  of  a  square 
that  shall  contain  12  acres  ? 

6.  A  company  of  speculators  bought  a  tract  of  land-  for 
$6724,  each  agreeing  to  pay  as  many  dollars  as  there  were 
partners  :   how  many  partners  were  there  ? 

7.  A  farmer  wishes  to  set  out  an  orchard  of  3844  trees, 
so  that  the  number  of  rows  shall  be  equal  to  the  number  of 
trees  in  each  row :'  what  will  be  the  number  of  trees  ? 

8.  How  many  rods  of  fence  will  inclose  a  square  field  of 
1 0  acres  ? 

9.  If  a  line  150  feet  long,  will  reach  from  the  top  of  a 
steeple    120   feet  high,   to   the   opposite   side   of  the   street, 
what  is  the  width  of  the  street  ? 

10.  What  is  the  length  of  a  brace  whose  ends  are  each 
3|  feet  from  the  angle  made  by  the  post  and  beam  ? 


CUBE   ROOT. 


CUBE    ROOT. 

369.  The  CUBE  ROOT  of  a  number  is  one  of  three  equal 
factors  of  the  number. 

To  extract  the  cube  root  of  a  number,  is  to  find  a  factor 
which,  multiplied  into  itself  twice,  will  produce  the  given 
number. 

Thus,  2  is  the  cube  root  of  8  ;  for,  2x2x2  =  8:  and  3 
is  the  cube  root  of  27  ;  for,  3x3x3  =  27. 

1,         2,         3,         4,         5,         6,         7,         8,         9. 
1          8         27        64       125      216      343      512      729 

The  numbers  in  the  first  line  are  the  cube  roots  of  the 
corresponding  numbers  of  the  second. 

370.  A  PERFECT  CUBE  is  a  number  which  has  three  exact 
factors.     By  examining  the  numbers  in  the  two  lines,  we  see, 

1st,  That  the  cube  of  units  cannot  give  a  higher  order 
than  hundreds. 

2d,  That  since  the  cube  of  one  ten  (10)  is  1000,  and  the 
cube  of  9  tens  (90),  81000,  the  oube  of  tens  will  not  give 
a  lower  denomination  than  thousands,  nor  a  higher  denomi- 
nation than  hundreds  of  thousands. 

Hence,  if  a  number  contains  more  than  three  figures,  its 
cube  root  will  contain  more  than  one  :  if  it  contains  more 
than  six,  its  root  will  contain  more  than  two,  and  so  on ; 
every  additional  three  figures  giving  one  additional  figure  in 
the  root ;  and  the  figures  which  remain  at  the  left  hand, 
although  less  than  three,  will  also  give  a  figure  in  the  root. 
This  law  explains  the  reason  for  pointing  off  into  periods  of 
three  figures  each. 

371.  Let  us  now  see  how  the  cube  of  any  number,  as  16, 
is  formed.     Sixteen  is  composed  of  1  ten  and  6  units,  and 
may  be  written  10  +  6.     To  find  the  cube  of  16,  or  of  10  +  6, 
we  must  multiply  the  number  bv  itself  twice. 

13 " 


290  EVOLUTION. 

To  do  this,  we  place  the  number  thus, 

Product  by  the  units, 
Product  by  the  tens, 


16  =  10+     6 
10+     fl 
60+   36 
100+     60 


Square  of  16,        .  .         .     100+   120+   36 

Multiply  again  by  16,  .  .         .                   10+     6 

Product  by  the  units,  .         .                       600+72,0+216 

Product  by  the  tens,  .         .  1000  +  1200+  360 

Cube  of  16,          ...         1000  +  180.0  +  1080+216 

1.  By  examining  the  parts  of  this  number,  it  is  seen  that 
the  first  part,  1000,  is  the  cube  of  the  tens;  that  is, 

10  x  10  x  10  =  1000. 

2.  The  second  part,  1800,  is  three  times  the  square  of  tft& 
tens  multiplied  by  the  units;  that  is, 

3  X  (10)2  X6^3xl00x6  =  1800. 

3.  The  third  part,  1080,  is  three  times  the  square  of  tike 
units  multiplied  by  the  tens;  that  is, 

3  x  62  x  10  =  3  x  36  x  10  =  1080. 

4.  The  fourth  part  is  the  cube  of  the  units;  that  is, 

63=  6  x  6  X  6=  216, 
1.  What  is  the  cube  root  of  the  number  4096? 

OPEKATION. 


4 
1 

12X3_=3)3 
163  —   4 


096(16 

0     (9-8-T-6 
096. 


ANALYSIS. — Since  the  number 
contains  more  than  three  figures, 
we  know  that  the  root  will  con- 
tain at  least  Ttnits  and  tens. 

Separating  the  three  right- 
hand  figures  from  the  4,  we 
know  that  the  cuhe  of  the  tens 
will  be  found  in  the  4;  and  1  is  the  greatest  cube  in  4. 

Hence,  we  place  the  root  1  on  the  right,  and  this  is  the  tens 
of  the  required  root.  We  then  cube  1,  and  subtract  the  result- 
from  the  first  period  4,  and  to  the  remainder  we  bring  down  the 
first  figure,  0,  of  the  next  period. 

We  have  seen  that  the  second  part  of  the  cube  of  16,  viz., 


CUBB    ROOT.  291 

is  thret  times  the  square  of  the  tens  multiplied,  ly  the  units;  and 
hence,  it  can  have  no  significant  figure  of  a  less  denomination  than 
hundreds.  It  must,  therefore,  make  up  a  part  of  the  30  hundreds 
above.  But  this  80  hundreds  also  contains  all  the  hundreds  which 
come  from  the  third  and  fourth  parts  of  the  cube  of  16.  Jf  it 
were  not  so,  the  30  hundreds,  divided  by  three  times  the  square 
of  the  tens,  would  give  the  unit  figure  exactly. 

Forming  a  divisor  of  three  times  the  square  of  the  tens,  we  find 
the  quotient  to  be  ten ;  but  this  we  know  to  be  too  large.  Placing 
9  in  the  root  and  cubing  19,  we  find  the  result  to  be  C859.  Then 
trying  8,  we  find  the  cube  of  8  still  too  large ;  but  when  we  take 
6,  we  find  the  exact  number.  Hence,  the  cube  root  of  4096  is  16. 

372.   Hence,  to  find  the  cube  root  of  a  number, 

Rule. — I.  Separate  the  given  number  into  periods  of  three 
figures  each,  by  placing  a  dot  over  the  place  of  units,  a 
second  over  the  place  of  thousands,  and  so  on  over  each  third 
figure  to  the  left;  the  left-hand  period  will  often  contain  less 
than  three  places  of  figures: 

II.  Note  the  greatest  perfect  cube  in  the  first  period,  and 
set  its  root  on  the  right,  after  the  manner  of  a  quotient  in 
division.     Subtract  the  cube  of  this  number  from  the  first 
period,  and  to  the  remainder  bring  down  the  first  figure  of 
the  next  period  for  a  dividend: 

III.  Take  three  times  the  square  of  the  root  just  found 
for  a  trial  divisor,  and  see  how  often  it  is  contained  in  the 
dividend,  and  place  the  quotient  for  a  second  figure  of  the 
root.     Then  cube  the  figures  of  the  root  thus  found ;  and  if 
their  cube  be  greater  than  the  first  two  periods  of  the  given 
number,  diminish  the  last  figure;  but  if  it  be  less,  subtract  it 
from  the  first  two  periods,  and  to  the  remainder  bring  down 
the  first  figure  of  the  next  period  for  a  new  dividend : 

IY.  Take  three  times  the  square  of  the  whole  root  for  a 
second  trial  divisor,  and  find  a  third  figure  of  the  root. 
(Jabe  the  whole  root  thus  found,  and  subtract  the  result  from 
the  first  three  periods  of  the  given  number  when  it  is  less 
than  that  number;  but  if  it  is  greater,  diminish  the  figure 
of  the  root :  proceed  in  a  similar  way  for  all  the  periods. 


292 


EVOLUTION. 


Examples. 

1.  What  is  the  cube  root  of  99252847  ? 

OPITRATION. 

99  252  847  (463 
43  =  64 

42  x  3  =  48  )  352,    dividend. 

First  two  periods,     .       .      99  252 
(46)3  =  46  x  46  x  46  =      97  336 

3  x  (46)2  =  6348)    19168,     2d  dividend. 

The  first  three  periods,    .      99  252  847 
(463)3  =       99  252  847 

Find  the  cube  roots  of  the  following  numbers : 


1.  Of  389017? 

2.  Of  5735339? 

3.  Of  32461759? 


Of   84604519? 
Of  259694072? 
Of  48228544? 


373.    To  extract  the  cube  root  of  a  decimal  fraction. 

Rule. — Annex  ciphers  to  the  decimal,  if  necessary,  so 
that  it  shall  consist  of  3,  6,  9,  &c.,  places.  Then  put  ike 
first  point  over  the  place  of  thousandths,  the  second  over 
the  place  of  millionths,  and  so  on  over  every  third  place  to 
the  right ;  after  which,  extract  the  root  as  in  whole  numbers. 

NOTES. — 1.  There  will  be  as  many  decimal  places  in  the  root, 
as  there  are  periods  in  the  given  number. 

2.  The  same  rule  applies,  when  the  given  number  is  composed 
of  a  whole  number  and  a  decimal. 

3.  If,  in  extracting  the  root  of  a  number,  there  is  a  remainder 
after  all  the  periods  have  been  brought  down,  periods  of  ciphers 
may  be  annexed,  by  considering  them  as  decimals. 

Examples. 

Find  the  cube  roots  of  the  following  numbers : 


1.  Of  .157464? 

2.  Of  .870983875? 

3.  Of  12.977875? 


4.  Of  .751089429? 

5.  Of  .353393243? 

6.  Of   3.408862625? 


CUBE   BOOT. 

374.    To  extract  the  cube  •  root  of  a  common  fraction. 

Rule. — I.  Reduce  compound  fractions  to  simple  ones, 
mixed  numbers  to  -improper  fractions,  and  then  reduce  the 
fraction  to  its  lowest  terms: 

II.  Extract  the  cube  root  of  the  numerator  and  denomi 
nator  separately,  if  they  have  exact  roots;  but  if  either  of 
them  has  not  cm  exact  root,  reduce  the  fraction  to  a  deci- 
mal and  extract  the  root  as  in  the  last  case. 

Examples. 

Find  the  cube  roots  of  the  following  fractions : 


1.  Of 

2.  Of 

3.  Of 


4.  Of  f  ? 

5.  Of  !•? 

6.  Of  f? 


Applications. 

1.  What  must  be  the  length,  depth,  and  breadth  of  a  box, 
when  these  dimensions  are  all  equal,  and  the  box  contains 
4913  cubic  feet  ? 

2.  The  solidity  of  a  cubical  block  is  21952  cubic  yards : 
what  is  the  length  of  each  side  ?     What  is  the  area  of  the 
surface  ? 

3.  A  cellar  is   25   feet  long,   20   feet  wide,   and   8|   feet 
deep :  what  will  be  the  dimensions  of  another  cellar  of  equal 
capacity,  in  the  form  of  a  cube  ? 

4.  What  will  be  the  length  of  one  side  of  a  cubical  gran- 
ary that  shall  contain  2500  bushels  of  grain  ? 

5.  How  many  small  cubes,  of  2  inches  on  a  side,  can  be 
sawed  out  of  a  cube  2  feet  on  a  side,  if  nothing  is  lost  in 
sawing  ? 

6.  What  will  be  the  side  of  a  cube  that  shall  be  equal  to 
the  contents  of  a  stick  of  timber  containing  1128  cubic  feet? 

7.  A  stick  of  timber  is  54  feet  long,  and  2  feet  square: 
what  would  be  its  dimensions,  if  it  had  the  form  of  a  cube  ? 


294 


ARITHMETICAL    PROGRESSION. 


NOTES. — 1.  Bodies  are  said. to  be  similar,  when  their  like  parts 
are  proportional  ? 

2.  It  is  found  that  the  contents  of  similar  bodies  are  to  each 
other  as  the  cubes  of  their  like  dimensions. 

3.  All  bodies  named  in  the  examples  below,  are  supposed  to 
be  similar. 

8.  If  a  sphere  of  4  feet  in  diameter  contains  33.5104  cu- 
bic  feet,  what  will  be  the  contents   of  a   sphere  §   feet  in 
diameter  ? 

4s    :     83    ::     33.5104    :     Ans. 

9.  If  the  contents  of  a  sphere  14  inches  in  diameter  is 
1436.7584  cubic  inches,  what  will  be  the  diameter  of  a  sphere 
which  contains  11494.0672  cubic  inches  ? 

10.  If  a  ball  weighing  32  pounds  is  6  inches  in  diameter, 
what  will  be  the  diameter  of  a  ball  weighing  2048  pounds  ? 

11.  If  a  haystack  24  feet  in  height,  contains  8  tons  of 
hay,  what  will  be  the  height  of  a  similar  stack  that  shall 
contain  but  1  ton  ? 


ARITHMETICAL  PROGRESSION. 

375.  An  ARITHMETICAL  PROGRESSION  is  a  series  of  numbers 
in  which,  each  is  derived  from  the  preceding  one,  by  the  ad- 
dition or  subtraction  of  the  same  number. 

The  number  added  or   subtracted  is  called,  the   common 
difference. 

376.  If  the  common  difference  is  added,  the  series  is  called, 
an  increasing  series. 

Thus,  if  we  begin  with  2,  and  add  the  common  difference 
3,  we  have, 

2,     5,     8,     11,     14,     17,     20,     23,   &c., 
which  is  an  increasing  series. 

If  we  begin  with  23,  and  subtract  the  common  difference 
3,  we  have, 

23,     20,     17,     14,     11,     8,     5,   &c, 
which  is  a  decreasing  series. 


ARITHMETICAL    PROGRESSION.  295 

The  several  numbers  arc  called,  the  terms  of  the  progres- 
sion or  series  :  the  first  and  last  are  called,  the  extremes, 
and  the  intermediate  terms  are  called,  means. 

377.  In  every  arithmetical  progression,  there  are  .five  parts  : 

1st,  The  first  term  ; 

2d,  The  last  term- 

3d,  The  common  difference; 

4th,  The  number  of  terms  ; 

5th,  The  sum  of  all  the  terms. 

If  any  three  of  these  parts  are  known  or  given,  the  re* 
reaming  ones  can  be  determined. 

CASE    I. 

378.  Knowing  the  first  term,  the  common  difference,  and 
the  number  of  terms,  to  find  the  last  term. 

1.  The  first  term  is  3,  the  common  difference  2,  and  the 
number  of  terms  19  :  what  is  the  last  term  ?  ^ 

• 

ANALYSIS.  —  By  considering  the  manner  in 

which  the  increasing  progression  is  formed,  OPERATION. 

we   see  that  the   2d   term   is   obtained  by  18,  No.  less  1. 

adding  the   common   difference   to   the  1st  2,  com.  dif. 

term  ;  the  3d,  by  adding  the  common  differ-  77 

dnce   to   the   2d;    the  4th,   by  adding  the  ^ 

comtnon   difference   to  the   3d,  and   so  on;  _' 

the  number  of  additions  being  1   less   than  39,  last  term. 
the  number  of  terms  found. 

But  instead  of  making  the  additions,  we  may  multiply  the 
common  difference  by  the  number  of  additions,  that  is,  by  1  less 
than,  the  number  of  terms,  and  add  the  first  term  to  the  product: 


Hiile.  —  Multiply  the  common  difference  by  1  less  than 
the  number  of  terms  ;  if  the  progression  is  increasing, 
add  the  product  to  the  first  term,  and  the  sum  will  be  the 
kwt  term  ;  if  it  is  decreasing,  subtract  the  product  from 
the  first  term,  and  the  difference  will  be  the  last  term. 


296 


ARITHMETICAL    PROGRESSION. 


Examples. 

1.  A  man  bought  50  yards  of  cloth,  for  which  he  was  to 
pay  6  cents  for  the  1st  yard,  9  cents  for  the  2d,  12  cents 
for  the  3d,  and  so   on,  increasing  by  the  common  difference 
3 :  how  much  did  he  pay  for  the  last  yard  ? 

2.  A  man  puts  out  $100  at  simple  interest,  at  7  per  cent.; 
at  the  end  of  the  1st  year  it  will  have  increased  to  $107,  at 
the  end  of  the  2d  year  to  $114,  and  so  on,  increasing  $7  each 
year:  what  will  be  the  amount  at  the  end  of  16  years  ? 

3.  What  is  the  40th  term  of  an  arithmetical  progression, 
of  which  the  first  term  is  1,  and  the  common  difference  1  ? 

4.  What  is  the  30th  term  of  a  descending  progression,  of 
which  the  first  term  is  60,  and  the  common  difference  2? 

5.  A  person  had  35  children  and  grandchildren,  and  it  so 
happened  that  the  difference  of  their  ages  was   18  months, 
and  the  age  of  the  eldest  was  60  years :   how  old  was  the 
youngest  ? 

CASE    II. 

379.  Knowing  the  two  extremes  and  the  number  of  terms, 
to  find  the  common  difference. 

1.  The  extremes  of  an  arithmetical  progression  are  8  and 
104,  and  the  number  of  terms  25  :  what  is  the  common  dif- 
ference ? 


OPERATION. 

104 
8 

-  1  =  24  )  96  (  4 


ANALYSIS. — Since  the  common  differ- 
ence multiplied  by  1  less  than  the  num- 
ber of  terms,  gives  a  product  equal  to 
the  difference  of  the  extremes,  if  we 
divide  the  difference  of  the  extremes  by 
1  less  than  the  number  of  terms,  the 
quotient  will  be  the  common  difference : 
Hence, 

Rule. — /Subtract  the  less  extreme  from  the  greater,  and 
divide  the  remainder  by  1  less  than  the  number  of  terms  ; 
the  quotient  will  be  the  common  difference. 


ARITHMETICAL    PROGRESSION.  597 

Examples. 

1.  A  man  has  8  sons  ;  the  youngest  is  4  years  old,  and  the 
eldest  32 :    their  ages   increase   in    arithmetical   progression : 
what  is  the  common  difference  of  their  ages  ? 

2.  A  man  is  to  travel  from  New  York  to  a  certain  place 
in  1 2  days ;  to  go  3  miles  the  first  day,  increasing  every  day 
by  the  same  number  of  miles  ;   the  last  day's  journey  is  58 
miles :  required  the  daily  increase. 

3.  A  man  hired  a  workman  for  a  month  of  26  working 
days,  and  agreed  to  pay  him  50  cents  for  the  first  day,  with 
a  uniform  daily  increase;   on  the   last  day  he   paid   $1.50: 
what  was  the  daily  increase  ? 

CASE    III. 

380.  To  find  the  sum  of  the  terms  of  an  arithmetical 
progression. 

1.  What  is  the  sum  of  the  series  whose  first  term  is  3, 
common  difference  2,  and  last  term  19  ? 

Given  series.        3+   5+   7  +   9  +  11  +  13  +  15  +  17  +  19—   99 

Same ;    order  ) 

of  terms  in- M9  +  17  +  15  +  13  +  11  +   9+   7+  5+   3-   99 
verted.  )  

Sum  of  both.     22    .22     22     22     22     22     22     22     22  =  198 

ANALYSIS. — The  two  series  are  the  same  ;  hence,  their  sum  is 
equal  to  twice  the  given  series.  But  their  sum  is  equal  to  the 
sum  of  the  two  extremes,  3  and  19,  taken  as  many  times  as  there 
are  terms ;  and  the  given  series  is  equal  to  half  this  sum,  or  to 
the  sum  of  the  extremes  multiplied  by  half  the  number  of  terms. 

Rule. — Add  the  extremes  together,  and  multiply  their 
sum  by  half  the  number  of  terms;  the  product  will  be  the 
sum  of  the  series. 

Examples. 

1.  The  extremes  are  2  and  100,  and  the  number  of  terrna 
22  :  what  is  the  sum  of  the  serie£  ? 

13* 


298 


GEOMETRICAL   PROGRESSION. 


ANALYSIS. — We  first  add 
together  the  two  extremes, 
and  then  multiply  by  half 
the  number  of  terms. 


OPERATION. 

2,    1st  term. 
100,    last  term. 

102,    sum  of  extremes. 
11,    half  the  number  of  terms. 


1122,    sum  of  series. 


2.  How  many  strokes  does  the  hammer  of  a  clock  strike 
in  12  hours  ? 

3.  The  first  term  of  a  series  is  2,  the  common  difference  4, 
and  the  number  of  terms  9 :  what  is  the  last  term  and  sum 
of  the  series  ? 

4.  James,  a  smart  chap,  having  learned  arithmetical  pro- 
gression, told  his  father  that  he  would  chop  a  load  of  wood 
of  15  logs,  at  2  cents  the  first  log,  with  a  regular  increase 
of  1  cent  for  each  additional  log :  how  much  did  James  re- 
ceive for  chopping  the  wood  ? 

5.  An  invalid  wishes  to'  gain  strength  by  regular  and  in- 
creasing exercise ;  his  physician  assures  him  that  he  can  walk 
1  mile  the  first  day,  and  increase  the  distance  half  a  mile  for 
each  of  the  24  following  days  :   how  far  will  he  walk  ? 

6.  If  100  eggs  are  placed  in  a  right  line,  exactly  one  yard 
from  each  other,  and  the  first  one  yard  from  a  basket :  what 
distance  will  a  man  travel  who  gathers  them  up  singly,  and 
places  them  in  the  basket? 


GEOMETRICAL    PROGRESSION. 

381.  A   GEOMETRICAL  PROGRESSION   is   a   series   of  terms, 
each  of  which  is  derived  from  the  preceding  one  by  multiply 
ing  it  by  a  constant  number.    The  constant  multiplier  is  called 
the  ratio  of  the  progression. 

382.  If  the  ratio  is  greater  than  1,  each  term  is  greater  than 
the  preceding  one,  and  the  series  is  said  to  be  increasing. 


GEOMETRICAL   PROGRESSION.  299 

If  the  ratio  is  less  than  1,  each  term  is  less  than  the  pre- 
ceding one,  and  the  series  is  said  to  be  decreasing;  thus, 
1,     2,     4,     8,  16,  32,  &c. — ratio  2 — increasing  series: 

82,  16,    8,     4,    2,     1,    &c. — ratio  \ — decreasing  series. 

The  several  numbers  are  called  terms  of  the  progression. 
The  first  and  last  are  called  the  extremes,  and  the  intermedi- 
ate terms  are  called  means. 

383.  In  every  Geometrical,  as  well  as  in  every  Arithmetical 
Progression,  there  are  five  parts : 

1st,    The  first  term  ; 
2d,     The  last  term  ; 
3d,     The  common  ratio  ; 
4th,    The  number  of  terms  ; 
5th,    The  sum  of  all  the  terms. 

If  any  three  of  these  parts  are  known,  or  given,  the  remain- 
ing ones  can  be  determined. 

CASE    I. 

384.  Having  given  the  first  term,  the  ratio,  and  the  n^m- 
ber  of  terms,  to  find  the  last  term. 

1.  The  first  term  is  3,  and  the  ratio  2i  what  is  the  6th 
term? 

ANALYSIS.— The   sec-  OPERATION. 

ond  term  is  formed  by      2x2x2x2x2  =  25  =  32 
multiplying  the  first  term  3_  1st  term, 

by  the  ratio;  the  third 
term  by  multiplying  the 
second  term  by  the  ratio,  and  so  on :  the  number  of  multiplicators 
being  1  less  than  the  number  qf  terms. 

3  =  3,  1st  term, 

3x2  =  6,  2d    term, 

3  X  2  x  2  =  3  x  22  =  12,  3d    term, 

3  X  2  x  2  x  2  =  3  x  23  =  24,  4th  term,  <foc. 

Therefore,  the  last  term  is  equal  to  the  first  term  multi- 
plied by  the  ratio  raised  to  a  power  1  less  than  the  number 
of  terms. 


300  GEOMETRICAL    PROGRESSION. 

Rule. — Raise  the  ratio  to  a  poioer  whose  exponent  is  1 
less  than  the  number  of  terms,  and  then  multiply  this  power 
by  the  first  term. 

Examples. 

1.  The  first  term  of  a  decreasing  progression  is  192  ;  the 
ratio  J,  and  the  number  of  terms  7 :  what  is  the  last  term  ? 

NOTE.— The  6th  power  of  the  ratio,  (|),  is         -    OPERATION. 
ffV;  and  this,  multiplied  by  the  first  term "l 92,  (£)6  =  ^. 

gives  the  last  term,  3.        .  192  x-1-  =  3. 

2.  A  man  purchased  12  pears ;   he  was  to  pay  1  farthing 
for  the  first,  2  farthings  for  the  second,  4  for  the  third,  and 
so  on,  doubling  each  time  :  what  did  he  pay  for  the  last  ? 

3.  The  first  term  of  a  decreasing  progression  is  1024,  the 
ratio  | :  what  is  the  9th  term  ? 

4.  The  first  term  of  an  increasing  progression  is  4,  and  the 
common  difference  3  :  what  is  the  10th  term? 

5.  A  gentleman  dying,  left  nine  sons,  and  bequeathed  his 
estate  in  the  following  manner :  To  his  executors,  $50  ;  to  his 
youngest  son  twice  as  much  as  the  executors,  and  each  other 
son*  double  the  amount  of  the  son  next  younger  :  what  was  the 
•eldest  son's  portion  ? 

6.  A  man  bought  12  yards  of  cloth,  giving  3  cents  for  the 
first  yard,  6  for  the  second,  12  for  the  third,  &c. :  what  did 
he  pay  for  the  last  yard? 

CASE    \l. 

385.  Knowing  the  two  extremes  and  the  ratio,  to  find 
the  sum  of  the  terms. 

1.  What  is  the  sum  of  the  terms,  in  the  progression  1,  4, 
16,  64? 

ANALYSIS. — If  we  multi-  OPERATION. 

ply  the  terms  of  the  pro-  4  +  16  +  64  +  256=       4  times 

gression  by  the  ratio  4,  we 

have  a  second  progression,     1+4  +  16  +  64  =  once. 

4,  16,  64,  256,  which  is  4  25^—1  =  3  times. 

times  as  great  as  the  first. 

If  from  this  we  subtract  the  256  ~  *  _  ?&>  _  g5 

first,  the  remainder,  256  — 1,  3  3 


GEOMETRICAL   PROGRESSIOjfc     >v  201 

will  be  3  times  as  great  as  the  first;  and  if  the  remainder  he 
divided  by  3,  the  quotient  will  be  the  sum  of  the  terms  of  the 
first  progression.  But  256  is  the  product  of  the  last  term  of  the 
given  progression  multiplied  by  the  ratio,  1  is  the  first  term,  and 
the  divisor,  3,  is  1  less  than  the  ratio:  Hence, 

Rule. — Multiply  the  last  term  by  the  ratio;  take  the  dif- 
ference between  the  product  and  the  first  term,  and  divide 
the  remainder  by  the  difference  between  1  and  the  ratio. 

NOTE. — When  the  progression  is  increasing,  the  first  term  is 
subtracted  from  the  product  of  the  last  term  by  the  ratio,  and 
the  divisor  is  found  by  subtracting  1  from  the  ratio.  When  the 
progression  is  decreasing,  the  product  of  the  last  term  by  the  ratio 
is  subtracted  from  the  first  term,  and  the  ratio  is  subtracted  from  1. 

Examples. 

1.  The  first  term  of  a  progression  is  2,  the  ratio  3,  and 
the  last  term  4375 :  what  is  the  sum  of  the  terms  ? 

2.  The  first  term  of  a  progression  is  12^,  the  ratio  \,  and 
and  the  last  term.  2 :  what  is  the  sum  of  the  terms  ? 

3.  The  first  term  is  3,  the  ratio  2,  and  the  last  term  192: 
what  is  the  sum  of  the  series  ? 

4.  A  gentleman  gave  his  daughter  in  marriage   on   New 
Year's  day,  and  gave   her  husband  Is.  toward  her  portion, 
and  was  to  double  it  on  the  first  day  of  every  month  during 
the  year :  what  was  her  portion  ? 

5.  A  man  bought  10  bushels  of  wheat,  on  the  condition 
that  he  should  pay  1  cent  for  the  1st  bushel,  3  foj-  the  2d, 
9  for  the  3d,  and  so  on  to  the  last :   what  did  he  pay  for 
the  last  bushel,  and  for  the  10  bushels  ? 

6.  A  man  has  6  children:  to  the  youngest  he  gives  $150  ; 
to  the  2d,  8300  ;  to  the  3d,  $600,  and  so  on,  to  each  twice 
as  much  as  to  the  one  before :  how  much  did  the  eldest  re- 
ceive, and  what  was  the  amount  received  by  them  all  ? 


302 


MENSURATION. 


D    a 


MENSURATION. 

386.  A  TRIANGLE  is  a  portion  of  a  plane,  bounded  by  three 
straight  lines.     It  has  thfiee  sides  and,  three  angles. 

BC  is  called,  the  base;  and  AD,  per- 
pendicular  to  BC,  the  altitude. 

387.  To  find  the  area  of  a  triangle. 

Rule. — Multiply  the  base  by  half  the 
altitude,  and  the  product  will  be  the  area. 
(Bk.  IV.  Prop.  VI.)*  B 

Examples. 

1.  The  base  BG,  of  a  triangle,  is  40  yards,  and  the  per- 
pendicular AD,  20  yards :  what  is  the  area  ? 

2.  In   a  triangular  field,  the  base  is  40  chains,  and  the 
perpendicular  15  chains  :  how  much  does  it  contain  ?    (Art. 
194.) 

3.  There  is  a  triangular  field,   of  which  the   base  is  35 
rods,  and  the  perpendicular  26  rods :  what  are  its  contents  ? 


388.  A   SQUARE   is   a  figure  haying  four 
equal  sides,  and  all  its  angles  right  angles. 

389.  A  RECTANGLE  is  a  four-sided  figure, 
like  a  square,  in  which  the  sides  are  perpen- 
dicular to  each  other,  but  the  adjacent  sides 
are  not  equal. 

390.  A    PARALLELOGRAM    is    a    four-sided 
figure  which  has  its  opposite  sides  equal  and 
parallel,  but  its  angles  not  right  angles. 

The"  line   DE,  perpendicular  to   the  base, 
is  called,  the  altitude. 


*  Davies'  Legendre. 


MENSURATION.  303 

391.    To  find  the   area  of  a  square,   rectangle,   or  paral- 
lelogram. 

Rule. — Multiply  the  base  by  the  perpendicular  height, 
and  the  product  will  *be  the  area.     (Bk.  IV.,  Prop.  V.) 

Examples. 

1.  What  is  the  area  of  a  square  field,  of  which  the  sides 
are  each  33.08  chains  ? 

2.  What  is  the  area  of  a  square  piece  of  land,  of  which 
the  sides  are  27  chains  ? 

3.  What  is  the  area,  of  a  square  piece  of,  land,  of  which 
the  sides  are  25  rods  each  ? 

4.  What  are  the  contents  of  a  rectangular  field,  the  length 
of  which  is  40  rods,  and  the  breadth  20  rods  ? 

5.  What  are  the  contents  of  a  field  40  rods  square  ? 

6.  What  are  the  contents  of  a  rectangular  field.  15  chains 
long,  and  5  chains  broad  ? 

7.  How  many  acres  in  a  field  27  chains  long  and  69  rods 
broad  ? 

8.  The  base  of  a  parallelogram  is  271  yards,  and  the  per- 
pendicular height  360  feet :  what  is  the  area  ? 


E  B 


392.  A  TRAPEZOID  is  a  four-sided  figure,  J) Q 

ABCD,  having  two  of  its  sides,  AB,  DC, 

parallel.     The  perpendicular,  CE,  is  called,  L — 
the  altitude. 

393.    To  find  the  area  of  a  trapezoid. 

Rule. — Multiply  half  the  sum  of  the  two  parallel  lines 
by  the  altitude,  and  the  product  luill  be  the  area.  (Bk.  IV., 
Prop.  VII.) 

Examples. 

1.  Required  the  area  of  the  trapezoid  ABCD,  having  given 
AB  =  321.51  ft,     DC  =  214.24  ft.,     and  ~  CE  =  171.16  ft. 

2.  What  is  the  area  of  a  trapezoid,  the  parallel  sides  of 
which  are  12.41  and  8.22  chains,  and  the  perpendicular  dis- 
tance between  them  5.15  chains  ? 


304  MENSURATION. 

3.  Required  the  area  of  a   trapezoid  whose  parallel  sides 
are  25  feet  6  inches,  and  18  feet  9  inches,   and  the  perpen- 
dicular distance  between  them  10  feet  and  5  inches. 

4.  Required  the  area  of  a  trapezoid  whose  parallel  sides 
are  20.5  and  12.25,  and  the  perpendicular  distance  between 
them  10.75  yards. 

5.  What  is  the   area  of  a  trapezoid  whose  parallel  sides 
are    7.50    chains    and    12.25    chains,    arid   the   perpendicular 
height  15.40  chains  ? 

6.  What  are  the  contents,  when  the  parallel  sides  are  20 
and  32  chains,  and  the  perpendicular  distance  between  them 
26  chains  ? 

394.  A  CIRCLE  is  a  portion  of  a  plane, 
bounded  by  a  curved  line,  called  the  cir- 
cumference. Every  point  of  the  circum- 
ference is  equally  distant  from  a  certain 
point  within,  called  the  centre:  thus,  C  is 
the  centre,  and  any  line,  as  ACB,  passing 
through  the  centre,  is  called,  a  diameter. 

If  the  diameter  of  a  circle  is  1,  the  circumference  will  be 
3.1416.  Hence,  if  we  know  the  diameter,  we  may  find  the 
circumference  by  multiplying  by  3.1416;  or,  if  we  know 
the  circumference,  we  may  find  the  diameter  by  dividing 
by  3.1416. 

.    Examples. 

1.  The  diameter  of  a  circle  is  4  :  what  is  the  circumference  ? 

2.  The  diameter  of  a  circle  is  93  :  what  is  the  circumference  ? 

3.  The  diameter  of  a  circle  is  20  :  what  is  the  circumference  ? 

4.  What  is  the  diameter  of  a  circle  whose  circumf.  is  78.54  ? 

5.  What  is  the  diameter  of  a  circle  whose  circumference  is 
11652.1944? 

6.  What  is  the  diameter  of  a  circle  whose  circumf.  is  6850  ? 

395.    To  find  the  area  or  contents  of  a  circle. 

Rule. — Multiply  the  square  of  the  radius  by  the  decimal, 
3.1416.     (Bk.  V.,  Prop.  XIV.,   Cor.  2.) 


MENSURATION. 


805 


G? 
10? 
7? 
Sift? 


Examples. 

1.  What  is  the  area  of  a  circle  whose  diameter  is 

2.  What  is  the  area  of  a  circle  whose  diameter  is 

3.  What  is  the  area  of  a  circle  whose  diameter  is 

4.  How  many  square  yards  in  a  circle  whose  diam.  is 

A 

396.  A  SPHERE  is  a  figure  bounded 

by  a  curved  surface,  all  the  parts  of 
which  are  equally  distant  from  a  cer- 
tain point  within,  called  the  centre. 
The  line  AB,  passing  through  its 
centre  C,  is  called,  the  diameter  of 
the  sphere,  and  AC,  its  radius. 


397.  To  find  the  surface  of  a  sphere. 

Rule. — Multiply  the  square  of  the  diameter  by  3.1416. 
(Bk.  VIIL,  Prop.  X.,  Cor.) 

Examples. 

1.  What  is  the  surface  of  a  sphere  whose  diameter  is  12? 

2.  What  is  the  surface  of  a  sphere  whose  diameter  is  7  ? 

3.  Required  the  number  of  square  inches  in  the  surface  of 
a  sphere  whose  diameter  is  2  feet,  or  24  inches. 

4.  How  many  square  miles  on  the  earth's  surface,  suppos- 
ing it  a  sphere,  whose  diameter  is  7912  miles  ? 

398.  To  find  the  contents  of  a  sphere.  » 

Rule. — Multiply  the  surface  by  the  radius,  and  divide 
the  product  by  3 :  the  quotient  mill  be  the  contents.  (Bk. 
VIIL,  Prop.  XI Y.) 

Examples. 

1.  What  are  the  contents  of  a  sphere  whose  diameter  is  12  ? 

2.  What  are  the  contents  of  a  sphere  whose  diameter  is  4  ? 

3.  What  are  the  contents  of  a  sphere  whose  diam.  is  14  in.  ? 

4.  What  are  the  contents  of  a  sphere  whose  diam.  is  6  ft.  ? 


306  MENSURATION. 

399.  A  prism  is  a  figure  whose  ends  are  eqmal 
plane  figures,  and  whose  faces  are  parallelograms. 

The  sum  of  the  sides  which  bound  the  base  is 
called  the  perimeter  of  the  base ;  and  the  sum  of 
the  parallelograms  which  bound  the  solid,  is  called 
the  convex  surface. 

400.  To  find  the  convex  surface  of  a  right  prism. 

Rule. — Multiply  the  perimeter  of  the  base  by  the  perpen- 
dicular height,  and  the  product  will  be  the  convex  surface. 
Cafe.  VII.  Prop.  I.} 

Examples. 

1.  What  is  the  convex  surface  of  a  prism  whose  base  is 
bounded  by  five  equal  sides,  each  of  which  is  35  feet,  the  alti- 
tude being  26  feet  ? 

2.  What  is  the  convex  surface  when  there  are  eight  equal 
sides,  each  15  feet  in  length,  and  the  altitude  is  12  feet  ? 

401.   To  find  the  solid  contents  of  a  prism. 

Rule. — Multiply  the  area  of  the  base  by  the  altitude,  and 
the  product  will  be  the  contents.  (Bk.  VII.,  Prop.  XIV.) 

Examples. 

1.  What  are  the  contents  of  a  square  prism,  each  side  of 
the  square  which  forms  the  base  being  15,  and  the  altitude 
of  the  prism  20  feet  ? 

2.  What  are  the  contents  of  a  cube,  each  side  of  which  is 
24  inches  ? 

3.  How  many  cubic  feet  in  a  block  of  marble,  of  which  the 
length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and  height 
or  thickness  2  feet  6  inches  ? 

4.  How  many  gallons  of  water  will  a  cis-tern  contain,  whose 
dimensions  are  the  same  as  in  the  last  example  ? 

5.  Required  the  contents  of  a  triangular  prism  whose  height 
is  10  feet,  and  area  of  the  base  350  ? 


MENSURATION. 


402.  A  cylinder  is  a  figure  with  circular 
ends.  The  line  E  F  is  called  the  axis,  or 
altitude ;  and  the  circular  surface,  the 
convex  surface  of  the  cylinder. 


403.    To  find  the  convex  surface. 

Rule. — Multiply  the  circumference  of  the  base  by  the 
altitude,  and  the  product  will  be  the  convex  surface. 
(Bk.  YIIL,  Prop.  I.) 

Examples. 

1.  What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  20,  and  the  altitude  50  ? 

2.  What  is  the  convex  surface  of  a  cylinder,  whose  altitude 
is  14  feet,  and  the  circumference  of  its  base  8  feet  4  inches  ? 

3.  What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  30  inches,  and  altitude  5  feet  ? 

404.    To  find  the  contents  of  a  cylinder. 

Rule. — Multiply  the  area  of  the  base  by  the  altitude: 
the  product  will  be  the  contents.  (Bk.  VIII.,  Prop.  II.) 

Examples. 

1.  Required  the  contents  of  a  cylinder,  of  which  the  alti- 
tude is  12  feet,  and  the  diameter  of  the  base  15  feet? 

2.  What  are  the  contents  of  a  cylinder,  the  diameter  of 
whose  base  is  20,  and  the  altitude  29? 

3.  How  many  barrels  of  wine  will  a  cylindrical  vat  fill,  the 
diameter  of  whose  base  is  12,  and^the  altitude  30? 

4.  What  are  the  contents,  in  hogsheads,  of  a  cylindrical 
cistern,  the  diameter  of  whose  base  is  16,  and  altitude  9? 

5.  What  are  the  contents  of  a  cylinder,  the  diameter  of 
whose  base  is  50,  and  altitude  15? 


808 


MENSURATION. 


405.  A  pyramid  is  a  figure  formed 
by  several  triangular  planes  united  at 
the  jtome  point  S,  and  terminating  in 
the  different  sides  of  a  plain  figure,  as 
ABODE.  The  altitude  of  the  pyra- 
mid is  the  line  S  0,  drawn  perpendicular 
to  the  base. 


406.    To  find  the  contents  of  a  pyramid. 

Rule. — Multiply  the  area  of  the  base  by  one-third  of  the 
altitude.    (Bk.  VIL,  Prop.  XVII.) 

Examples. 

1.  Required  the  contents  of  a  pyramid,  of  which  the  area 
of  the  base  is  95,  and  the  altitude  15  ? 

2.  What  are  the  contents  of  a  pyramid,  the  area  of  whose 
base  is  260,  and  the  altitude  24  ? 

3.  What  are  the  contents  of  a  pyramid,  the  area  of  whose 
base  is  207,  and  altitude  18  ? 

4.  What  are  the  contents  of  a  pyramid,  the  area  of  whose 
base  is  403,  and  altitude  36  ? 

5.  What  are  the  contents  of  a  pyramid,  the  area  of  whose 
base  is  270,  and  altitude  16? 

6.  A  pyramid  has  a  rectangular  base,  the  sides  of  which 
are  25  and  12  :  the  altitude  of  the  pyramid  is  36  :  what  are 
its  contents  ?  G 


407.  A  cone  is  a  figure  with  a  cir- 
cular base,  and  tapering  to  a  point 
called  the  vertex.  Tilie  point  C  is  the 
vertex,  and  the  line  C  B  is  called  the 
axis,  or  altitude. 


GAUGING.  SOO 

408.     To  find  the  contents  of  a  cone. 
Rule. — Multiply  the  area  of  the  base  by  the  altitude,  and 
divide  the  product  by  3.    (Bk.  VIII.,  Prop.  V.) 

Examples. 

1.  Required  the  contents  of  a  cone,  the  diameter  of  whose 
base  is  5,  and  the  altitude  10? 

2.  What  are  the  contents  of  a  cone,  the  diameter  of  whose 
base  is  18,  and  the  altitude  27  ? 

3.  What  are  the  contents  of  a  cone,  the  diameter  of  whose 
base  is  20,  and  the  altitude  30  ? 

4.  What  are  the  contents  of  a  cone,  whose  altitude  is  27 
feet,  and  the  diameter  of  the  base  10  feet  ? 

5.  What  are  the  contents  of  a  cone,  whose  altitude  is  12 
feet,  and  the  diameter  of  its  base  15  feet? 


GAUGING-. 

.409.  GAUGING  is  a  process  for  determining  the  capacity  or 
contents  of  casks. 

The  mean  diameter  of  a  cask  is  found  by  adding  to^  the 
head  diameter,  two-thirds  of  the  difference  between  the  bung 
and  head  diameters,  or,  if  the  staves  are  not  much  curved, 
by  adding  six-tenths.  This  reduces  the  cask  to  a  cylinder. 
Then,  to  find  the  contents,  we  multiply  the  square  of  the  mean 
diameter  by  the  decimal  .7854,  and  the  product  by  the  length. 
This  will  give  the  contents  in  cubic  inches.  Then,  if  we  divide 
by  231,  we  have  the  contents  in  gallons  (Art.  199). 

Multiply  the  length  by  the  square  of  the  OPERATION. 


mean  diameter,  then  by  the  decimal  .7854,  ^  X  d*  X  '^-—  := 
and  divide  by  281.  Z  x  d«  x!o034. 

If,   tlien,  we   duide   the   decimal   .7854 

by  231,  the  quotient,  carried  to  four  places  of  decimals,  is  .0034, 
and  this  decimal  multiplied  by  the  square  of  the  mean  diameter 
and  by  the  length  of  the  cask,  will  give  the  contents  in  gallons. 


310  GAUGING. 

410.  Hence,  for  gauging  or  measuring  casks,  we  have  the 
following 

Rule. — Multiply  the  length  by  the  square  of  the  mean 
diameter;  then  multiply  by  34,  and  point  off  four  decimal 
places,  and  the  product  will  then  express  gallons  and  the 
decimals  of  a  gallon. 

Examples. 

1.  How  many  gallons  in  a  cask  whose  bung  diameter  is  36 
inches,  head  diameter  30  inches,  and  length  50  inches  ? 

We  first  find  the  difference  of  the  diame-  OPEBATION. 

ters,  of  which  we  take  two-thirds,  and  add  36  —  30—6 

to  the  head  diameter.     We  then  multiply  the  2   Of  g  _    4 

square  of  the  mean  diameter,  the  length,  and  „  .           _  „ 
34  together,  and  point  off  four  decimal  places 

in  the  product.  =  1156 

1156  x  50  x  34  =  196.52  gal. 

2.  What  is  the  number  of  gallons  in  a  cask  whose  bung 
diameter  is  38  inches,  head  diameter  32  inches,  and  length 
42  inches  ? 

3.  How  many  gallons  in  a  cask  whose  length  is  36  inches, 
bung  diameter  35  inches,  and  head  diameter  30  inches  ? 

4.  How  many  gallons  in  a  cask  whose  length  is  40  inches, 
head  diameter  34  inches,  and  bung  diameter  38  inches  ? 

5.  A  water-tub  holds  147  gallons;  the  pipe  usually  brings 
in  14  gallons  in  9  minutes;  the  tap  discharges  at  a  medium, 
40  gallons  in  31  minutes.     Now,  supposing  these  to  be  left 
open,  and  the  water  to   be   turned  on  at   2   o'clock   in   the 
morning ;  a  servant  at  5  shuts  the  tap,  and  is  solicitous  to 
know  at  what  time  the  tub  will  be  filled,  in  case  the  water 
continues  to  flow. 


PROMISCUOUS   EXAMPLES.  311 


Promiscuous  Examples. 

1.  Sound  travels  about  1142  feet  in  a  second;  now,  if  the 
flash  of  a  cannon  is  seen  at  the  moment  it  is  fired,  and  the 
report  heard  45  seconds  after,  what  distance  would  the  ob- 
server be  from  the  gun  ? 

2.  Two  persons  depart  from  the  same  place ;  one  travels 
32,  and  the  other  36  miles  a  day :  if  they  travel  in  the  same 
direction,  how  far  will  they  be  apart  at  the  end  of  19  days, 
and  how  far,  if  they  travel  in  contrary  directions  ? 

3.  A  traveler  leaves  New  Haven  at  8  o'clock  on  Monday 
morning,  and  walks  toward  Albany,  at  the  rate  of  3  miles 
an  hour ;  another  traveler  sets  out  from  Albany  at  4  o'clock 
on  the  same  evening,  and  walks  toward  New  Haven,  at  the 
rate  of  4  miles  an  hour :   now,  supposing  the  distance  to  be 
130  miles,  where  on  the  road  will  they  meet  ? 

4.  Two  persons,  A  and  B,   are  indebted   to  C ;    A  owes 
$2173,   which  is   the   least   debt,    and   the   difference   of  the 
debts  is  $371:  what  is  the  amount  of  their  indebtedness? 

5.  What   number,   added   to  the  43d   part   of  4429,  will 
make  the  sum  240  ? 

6.  What  number  is  that  which,  being  multiplied  by  §,  will 
produce  J? 

7.  A  tailor  had  a  piece   of  cloth  containing  24 1   yards, 
from  which  he  cut  6 J  yards :   how  much  was  there  left  ? 

8.  From  f  of  ff ,  take  i  of  ^. 

12  o 

9.  What  is  the  difference  between  3f  +  7£,  and  4  +  2}-J? 

10.  The  product  of  two  mimbers  is  2.26,  and  one  of  the 
numbers  is  .25 :  what  is  the  other  ? 

11.  If  the  divisor  of  a  certain  number  be  6.66f,  and  the 
quotient  f ,  what  will  be  the  dividend  ? 

12.  A  merchant  bought  13  packages  of  goods,  for  which 
he  paid  $326  :  what  will  39  packages  cost,  at  the  same  rate  ? 

13.  How  many  bushels  of  oats,  at  62J  cents  a  bushel,  will 
pay  for  4250  feet  of  lumber,  at  $7.50  per  thousand  ? 

14.  Bought  2  hhd.  of  sugar,  which  weighed  as  follows :  th« 
1st,  5  cwt.  1  qr.  181b.;   the  2d,  6  cwt.  10  Ib. :   what  did  it 
cost,  at  7  cents  per  pound  ? 


313  PROMISCUOUS    EXAMPLES. 

15.  How  many  hours  between  the  4th  of  Sept.,  1854,  at 
3  P.  M.,  and  the  20th  day  of  April,  1855,  at  10  A.  M.  ? 

16.  If  |-  of  a  gallon  of  wine  costs  f  of  a  dollar,  what  will 
f  of  a  hogshead  cost  ? 

17.  The  sum  of  two  numbers  is  425,  and  their  difference 
1.625  :   what  are  the  numbers  ? 

18.  The  sum  of  two  numbers  is  |-,  and  their  difference  J: 
what  are  the  numbers  ? 

19.  What  is  the  difference  between  twice  five  and  fifty,  and 
twice  fifty-five  ? 

20.  What  number  is  that  which,  being  multiplied  by  three- 
thousandths,  the  product  will  be  2637  ? 

21.  What  is  the  difference  between  half  a   dozen  dozens 
and  six  dozen  dozens  ? 

22.  The  slow,  or  parade  step,  is  70  paces  per  minute,  at 
28  inches  each  pace :   how  fast  is  that  per  hour  ? 

23.  A  person  dying,  divided  his  property  between  his  widow 
and  his  four  sons  ;  to  his  widow  he  gave  $1780,  and  to  each 
of  his  sons,  $1250  ;  he  had  been  25|  years  in  business,  and 
had  cleared,  on  an  average,  $126  a  year:  how  much  had  he 
when  he  began  business  ? 

24.  How  many  planks,  15  feet  long  and  15  inches  wide, 
will  floor  a  barn,  60J  feet  long  and  33-*-  feet  wide  ? 

25.  A  room  30  feet  long  and  18  feet  wide,  is  to  be  cov- 
ered with  painted  cloth  f  of  a  yard  wide :  how  many  yards 
will  cover  it  ? 

26.  There  was  a  company  of  soldiers,  of  whom  ^  were  on 
guard,  i  preparing  dinner,  and  the  remainder,  85  men,  were 
drilling :    how  many  were  there  in  the  company  ? 

27.  A  person  owned  f  of  a  mine,  and  sold  f  of  his  inter- 
est for  $1710:  what  was  the  value  of  the  entire  mine? 

28.  In  a  certain  orchard,  |  of  the  trees  bear  apples,  \  of 
them  bear  peaches,  -J-  of  them  plums,   120  of  them  cherries, 
and   80   of  them  pears :    how  many  trees  are   there  in  the 
orchard  ? 

29.  A,  B,  and  C  trade  together,  and  gain  $120,  which  is 
to  be  shared  according  to  each  one's  stock ;  A  put  in  $140, 
B  $300,  and  C  $160:   what  is  each  man's  share? 

30.  Four  persons  traded  together,  on  a  capital  of  86000, 
of  which  A  put  in  J,  B  put  in  ^  C  put  in  £,  and  D  the 


PROMISCUOUS    EXAMPLES.  oil'. 

rest;    at  the  end  of  4  years,  they  had  gained  $4728:   what 
was  each  one's  share  of  the  gain  ? 

31.  A  can  do  a  piece,  of  work  in  12  days,  and  1>  can  do 
the  same  work  in  18  days:    how  long  will  it  take  both,  if 
they  work  together? 

32.  If  a  barrel  of  flour  will  last  one  family  7{,  months,  a 
second  family  9  months,  and  a  third  11 J  mouths,  how  long 
will  it  last  the  three  families  together  ? 

33.  Suppose  I  have  T3g-  of  a  ship  worth  $1200:  what  part 
have  I  left  after  selling  f  of  J  of  my  share,  and  what  is  it 
worth  ? 

34.  What  number  is  that  which,  being  multiplied  by  f  of 
|  Df  li,  the  product  will  be  1  ? 

5^5)  Divide  $420  among  three  persons,  so  that  the  second 
shall  have  j  as  much  as  the  first,  and  the  third  J  as  much 
as  the  other  two. 

36.  Divide  $10429.50  among  three  persons,  so  that  as  often 
as  one  gets  $4,  the  second  will  get  $6,  and  the  third  $7. 

37.  A  gentleman  whose  annual  income  is  .£1500,  spends  20 
guineas'  a  week  :  does  he  save,  or  run  in  debt,  arid  how  much  ? 

38.  A  lady  being  asked  her  age,  and  not  wishing  to  give 
a  direct  answer,,  said:  "I  ha/e  9  children,  and  three  years 
elapsed  between  the  birth  of  each  of  them;    the  eldest  was 
born  when   I  was   19  years   old,   and  the  youngest   is   now 
exactly  19 :"  what  was  her  age  ? 

39.  A  wall  of  700  yards  in  length,  was  to  be  built  in  29 
days;   12  men  were  employed  on  it   for   11  days,   and  only 
completed   220   yards  :    how   many  men  must  be   added,    to 
complete  the  wall  in  the  required  time  ? 

40.  A  besieged  garrison,  consisting  of  360  men,  was  pro- 
visioned for  6  months,  but  hearing  of  no  relief  at  the  end  of 
5  months,  dismissed  so  many  of  the  garrison,  that  the  remaining 
provisions  lasted  5  months :  how  many  men  were  sent  away  ? 

41.  A  farmer  exchanged  70  bushels  of  rye,  at  $0.92  per 
oushel,  for  40  bushels  of  wheat,  at  $1.37^  a  bushel,  and  re- 
ceived the  balance  in  oats,  at  $0.40  per  bushel :    how  many 
bushels  of  oats  did  he  receive  ? 

42.  If  a  quantity  of  provisions  serves  1500  men  12  weeks, 
at  the  rate  of  20  ounces  a  day  for  each  man,  how  many  men 
will  the  same  provisions  maintain  for  20  weeks,  at  the  rate 
of  8  ounces  a  day  for  each  man  ? 

14 


314:  PROMISCUOUS    EXAMPLES. 

43.  How  many  bricks,  8  inches   long  and  4,  inches  wide, 
will  pave  a  yard  that  is  100  feet  by  50  feet  ?  ' 

44.  How  many  stones,   2   feet  long,    1   foot  wide,   and   6 
inches  thick,  will  build  a  wall  12  yards  long,  2  yards  high, 
and  4  feet  thick? 

45.  If  20  men  perform  a  work  in  12  days,  how  many  men 
will  accomplish  thrice  as  much  in  one-fifth  of  the  time  ? 

46.  Twelve  workmen,  working  12  hours  a  day,  hare  made, 
in  12  days,  12  pieces  of  cloth,  each  piece    75   yards  long : 
how  many  pieces  of  the  same  stuff  would  have  been  made, 
each  piece  25  yards  long,  if  there  had  been  7  more  workmen  ? 

47.  A  person  was  born  on  the  1st  day  of  Oct.,  1801,  at 
6  o'clock  in  the  morning :  what  was  his  age  on  the  21st  of 
Sept.,   1854,  at  half-past  4  in  the  afternoon  ? 

48.  A  man  went  to  sea  at  17  years  of  age;  8  years  after, 
he  had  a  son  born,  who  lived  46  years,  and  died  before  his 
father;  after  which  the  father  lived  twice  twenty  years,  and 
died :  what  was  the  age  of  the  father  ? 

49.  A  can  do  a  piece  of  work,  alone,  in  10  days,  and  B 
in  13  days  :  in  what  time  can  they  do  it  if  they  work  together  ? 

50.  A  cistern,  containing   60  gallons   of  water,  has  three 
unequal  pipes  for  discharging  it ;  the  largest  will  empty  it  in 
one  hour,  the  second  in  two  hours,  and  the  third  in  three  hours  : 
in  what  time  will  the  cistern  be  emptied  if  they  run  together  ? 

51.  A  man  bought  f  of  the  capital  of  a  cotton  factory, 
at  par  ;  he  retained  £  of  his  purchase,  and  sold  the  balance 
for  $5000,  which  was  15  per  cent,  advance  on  the  cost :  what 
was  the  whole  capital  of  the  factory  ? 

52.  Bought  a  cow  for  $30  cash,  and  sold  her  for  $35  at 
a  credit  of  8  months :  reckoning  the  interest  at  6  per  cent., 
how  much  did  I  gain  ? 

53.  If,  when  I  sell  cloth  for  8s.  9d.  per  yard,  I  gain  12 
per  cent.,  what  per  cent,  will  be  gained  when  it  is  sold  for 
10s.  6d.  per  yard? 

54.  How  much  stock,  at  par  value,  can  be  purchased  for 
$8500,  at   8|  per  cent,  premium,   J  per  cent,  being  paid  to 
the  broker  ?  " 

55.  Divide  $500  among  4  persons,  so  that  when -A  has  -| 
of  a  dollar,  B  shall  have  J,  0,  |,  and  D,  }. 

56.  Three  persons  purchase  a  piece  of  property  for  $9202  ; 
the  first  gave  a  certain  sum,  the  second  three  times  as  much. 


PROMISCUOUS    EXAMPLES.  315 

and  the  third  one  and  a  half  times  as   much  as   the  other 
two  :   .what  did  each  pay  ? 

57.  A  ship  has  a  leak,  by  which  it  would  fill  and  sink  in 
15  hours,  but,  by  means  of  a  pump,  it  could  be  emptied,  if 
full,  in  16  hours.     Now;  if  the  pump  is  worked  from  the  time 
the  leak  begins,  how  long  before  the  ship  will  sink  ? 

58.  A  reservoir  of  water  has  two  cocks  to  supply  it ;  the 
first  would  fill  it  in  40  minutes,  and  the   second  in  50.     It 
has  likewise  a  discharging  cock,  by  which  it  may  be  emptied, 
when  full,  in  25  minutes.     Now,  if  all  the  cocks  are  opened 
at  once,  and  the  water  runs  uniformly  as  we  have  supposed, 
how  long  before  the  cistern  will  be  filled  ? 

59.  If  a  house  is  50  feet  wide,  and  the  post  which  sup- 
ports the  ridge-pole  is  12  feet  high,  what  will  be  the  length 
of  the  rafters  ? 

60.  A  man  had   12  sons;   the  youngest  was  3  years  old 
and  the  eldest  58,  and  their   ages   increased  in  arithmetical 
progression :  what  was  the  common  difference  of  their  ages  ? 

61.  A  man  bought  10  bushels  of  wheat,  on  the  condition 
that  he  should  pay  1  cent  for  the  1st  bushel,  3  for  the  2d, 
9  for  the  3d,  and  so  on  to  the  last :    what  did  he  pay  for 
the  last  bushel,  and  for  the  10  bushels  ? 

62.  There  is  a  mixture  made  of  wheat  at  4s.  per  bushel, 
rye  at  3s.,  barley  at  2s.,  with  12  bushels  of  oats  at  18d.  per 
bushel :   what  proportion  must  be  taken  of  each  sort,  to  make 
the  mixture  worth  3s.  6d.  per  bushel  ? 

63.  What  length  must  be  cut  off  a  board  8|  inches  broad, 
to  contain  a  square  foot  ? 

64.  What  is  the  difference  between  the  interest  of  $2500, 
for  4  years  9  months,  at  6  per  cent.,  and  half  that  sum  for 
twice  the  time,  at  half  the  same  rate  per  cent.? 

65.  A  person  lent  a  certain  sum  at  4  per  cent,  per  annum ; 
had  this  remained  at  interest  3  years,  he  would  have  received 
for  principal  and  interest,  $9676.80:  what  was  the  principal  ? 

66.  In  what  time  will  $2377.50  amount  to  $2852.42,  at  4 
per  cent,  per  annum  ? 

67.  A  man  purchased  a  building  lot,  containing  3600  square 
feet,  at  the  cost  of  $1.50  per  foot,  on  which  he  built  a  store 
at  an  expense  of  $3000.     He  paid  yearly  $180.66  for  repairs 
and  taxes :    what  annual  rent  must  he   receive,  to  obtain  10 
per  cent,  on  the  cost  ? 


316  PROMISCUOUS    EXAMPLES. 

68.  A's  note,  of  $7851.04,  was  dated  Sept.  5th,  1837,  on 
which  were  indorsed  the  following  payments,  viz. :  NOT.  13th, 
1839,  $416.98  ;  May  10th,  1840,  $152  :  what  was  chie  March 
1st,  1841,  the  interest  being  6  per  cent.  ? 

69.  If  1  pound  of  tea  be  equal  in  value  to  50  oranges, 
and  70  oranges  be  worth  84  lemons  :  what  is  the  value  of  a 
pound  of  tea,  when  a  lemon  is  worth  2  cents  ? 

70.  A  person  bought  160  oranges  at  2  for  a  penny,  and  180 
more  at  3  for  a  penny ;  after  which  he  sold  them  out  at  the 
rate  of  5  for  2  pence  :  did  he  make  or  lose,  and  how  much  ? 

71.  A  snail,  in  getting  up  a  pole  20  feet  high,  was  observed 
to  climb  up  8  feet  every  day,  but  to  descend  4  feet  every 
night :  in  what  time  did  he  rekch  the  top  of  the  pole  ? 

72.  What  is   the  height  of  a  wall,  which  is  14J  yards  in 
length,  and  T7g  of  a  yard  in  thickness,  and  which  has  cost  $406, 
it  having  been  paid  for  at  the  rate  of  $10  per  cubic  yard  ? 

73.  What  will  be  the  duty  on  225  bags   of  coffee,  each 
weighing  gross  160  Ibs.,  invoiced  at  6  cts.  per  Ib. ;  2  per  cent, 
being  the  legal  rate  of  tare,  and  20  per  cent,  the  duty  ? 

74.  A  house  is  40  feet  from  the  ground  to  the  eaves,  and  it 
is  required  to  find  the  length  of  a  ladder  which  will  reach  the 
eaves,  supposing  the  foot  of  the  ladder  cannot  be  placed  nearer 
to  the  house  than  30  feet  ? 

75.  A  person  dying,  worth  $5460,  left  a  wife  and  2  children, 
a  son  and  daughter,  absent  in  a  foreign  country.     He  directed 
that,  if  his  son  returned,  the  mother  should  have  one-third  of 
the  estate,  and  the  son  the  remainder ;  but  if  the  daughter 
returned,  she  should  have  one-third,  and  the  mother  the  re- 
mainder.    Now  it  so  happened  that  they  both  returned :  how 
must  the  estate  be  divided  to  fulfil  the  father's  intentions  ? 

76.  If  a  cylindrical  cistern,  8  feet  in  diameter,  will  hold  120 
barrels,  what  must  be  the  diameter  of  a  cistern  of  the  same 
depth,  to  hold  1500  barrels? 

77.  If  A  is  40  years  old  and  B  is  16,  how  many  years 
since  A  was  9  times  as  old  as  B  ? 

78.  A  can  earn  a  certain  sum  of  money  in  20  days  :  A  and 
B  together,  can  earn  the  same  sum  in  6  days  :  how  long  will 
it  take  B,  alone,  to  earn  the  same  sum  ? 

79.  A  and  B  can  perform  a  certain  piece  of  work  in  6  days, 
B  and  C  in  7  days,  and  A  and  C  in  14  days  :  in  what  time 
would  each  do  it  alone? 


ANSWERS. 


PAGE.      EX.      ANS.       F.X.      ANS.      KX.    ANS.     KX. 


17. 

1  |  30  ||  2  |  80  ||  3  9  ||  4  |  8654  |  5  90876  |  6  |  154970 

17. 

7  |  90813  1  8  780571  ||  9 

9023929  1  10  1  86981028 

17. 

11  |  1409060760503  |  12  | 

13800769703009  ||  13  | 

17. 

907400832106000201  ||  14  |  86981028  ||  15  |  168765432 

19. 

1  |  105  ||  2  j  302  ||  3  519  |  4 

1004  1  5  |  8701  ||  6  |  40406 

19.  ||  7  |  58061  ||  8  |  99999  ||  9 

|  406049  j  10  |  641721 

19.  ||  11  |  1421602  ||  12  |  9621016 

13  |  94807409  ||  14  |  4000- 

f97~||3  06  909    1  5  49000  000949  (JO  5  ||  1  0  1  1)00000000999990999 

1  97  II   17"|  409000000000000000209106  ||  2l7HT|  209"  ||  275005 
2irf~3  I   12012  ||  6  |  100101  ||  9  |  47204851  ||  10  |  6049072000- 
21.  1   407861  ||  11  j  899460850200506499  ||  12  |  590590o9059^ 

2  1  .  I   95  9  ||  1  3'|  1  2  1  1  1  1|  1  4  |  9000000065  ||  1  5  ~\3Q4000l  00  32  1  !M  1 


305104001045074TI 


21.  ||   80301006004620   |    26.   ||   1  |  196   ||   2  |  749   ||  3  |  1347 


26.  1   4  |  566  I  5  |  1183  ||  6     1397  |  7  |  999  ||  8  |  689  ||  9  |  9789 


26.  1    10      9799   ||    11    |    12089    ||    12    |    26901    ||    13    |    28637 


26-JU±_l_?039:33   I 
26.      18     1994439 


22     10742750 


15  |  23272   I  16  |  233642   ||  Jtf_|_24  7481 

19  |  175874~|P20  |  172775  ||~2iy98967 
~25~~787C76921    [    26    [    100570011 


27- 

27 

15371791930  ||  28  |  577  |  29  |  7689  ||  30  |  502616 

27. 

31 

799999  ||  32  |  43  ||  33  |  73  ||  34  j  888  ||  35  |  68-16 

28. 

36 

9798  1  37  |  8601  ||  38  |  7032  |  39  |  979  ||  40  |  559 

28. 

41 

26754  ||  42  ,  730528  ||  43  |  7047897  |  44  |  25687540 

45  |  297303078  ||  46  |  13115375  |  47  |  39428059  ||  48  | 
29.  ||   140700031HI  49      1819857171537  |  50  |  1105364  ||  51  | 


29.  ||  1079167  |  52  |  1118969  ||  53  |  1665400   ||   30.  ||  1  |  365 

30.  i  2  |  137   ||   3  |  4025   ||   4  |  2800   ||  5   |   5567   [   6  |  16375 

30.  -||  7  |  8  h.,  20  g.,  8  o.,  12  c.,  36  e.,  17  .?/.  c.,  320  6-.— all,  421 

31.  ||  8  |  392  ||  9  |  1880  ||  10  j  84237  ||  11  |  1100  |  12  |  4107 
3l7||  13  |T467~1660 "II  1 4  ]Tl2869  |  15  |  2576406  ||  16  |  370 


318 


ANSWERS,. 


32.  ||    17  |  1199-4596  ||  18  |  42390529-4530902  ||  19  |  1287462 


32.  ||  20 

50994  |  21  j  94341  ||  22  |  143985  ||  23  |  2728116 

32-1 

24  |  1862  1  35.  ||  2  |  632  ||  3  |  1010  ||  4  11123  ||  5  |  341 

36.  || 

6  |  111342  |  7  .  724221  ||  8  |  10403360  ||  9  |  17102 

36.|| 

10 

7513725  1  11  |  58185356  ||  12  |  114865607 

36-11 

13 

2774  ||  14  |  1737  ||  15  85991  j  16  |  27087  ||  17  |  0000 

36.  || 

18 

20001  1  19  |  99999  ||  20  |  1396765  ||  21  |  9949994 

36.  || 

22 

260822  1  23  |  2935621  |  24  |  50391719  ||  25  |  28443 

36.  || 

26 

99246591  ||  27  |  999999  ||  28  |  776462  ||  29  |  185- 

36.|| 

61747  ||  30  |  4244088  ||  31  |  8013105  ||  32  |  52528  ||  33  | 

56001996606  ||  34  |  140981 9896|  35  |709704201330 


37^ 

?Zi 

38.^ 

3j^ 
38. 
38. 
39^ 
40^ 
40._ 
40. 

44i 
4£ 

44. 


1115   ||   2     1894  ||   3  |  10  ||  4  |  45  ||  5  |  67  ||  6  |  62 

~ 


||   7     785608   ||   8 


9     5057   ||    10     3632|l   |  696 


12  |  1825  ||  13  |  1732  ||  14 


15  |  1759  ||  16  |  79 


175502  II  18  |  1860805  ||  19  |  239  ||  20  |  250-1500 


21  |  1340-4020  I  22  2769818  J  23  |  145  ||  24  7906 
" 2 :  5~ | Td^5"||T6"7 15~974260"|| "2 77  20463760J ~28  25579 
~2iT| "2276525"!  1  |  29045" f  2  |  418  J  3  \  77~4  ||  4  |  5795 

5  |  390   I   6  |  224980    |    7    |   230-527    |    8   |    19553068 


9   919  ||  10  |  55  ||  11  |  28223  ||  12  |  170G  |  13   3818 


14  |  1 1854617  ||  44.  ||  1  |  867901  ||  2  557808  ||  3  |  2030223 


4  |  191616  I  5  |  3903175  |  6  |  5462172 
8|  3  2~(T||~ir  |~2  :  7  144~  ||  1  0  |  536392  |  1  1  | 


||  7  12533346 
1  352060  1  2 


212912J  13  |  000000  ||  14  |  182982  ||  15  |  347535  ||  16  |  936 
lY~|T236~  |7 8  I  29^1  "19726766"  ||  20  |  28511  ||  21  |  5382 
22  |  1485~j j~467|T~ 30660  ||  2  |  7913576  ||  3  f7723C82 


44 

48 
46 

46.  ||  11  j  65948806||  12  |  369141  7.0  ||  13  |  85950000  ||  14  |  3320- 

4fr 
46 


4  |  4280822  ||  5  |  19014604  ||  6  [85564584  I  7   2183178497 
8  |  93969^64472    j    9    |   395061696    ||    10    |    393916488 


863272  ||  15  |  876515040  ||  16  |  68959498  ||  17  |  62415- 


97890  ||  18  |  105062176  ||  19  |  294360066  |j  20  |  498- 


|  155396  ||  21  j  406070736  ||  22   800105244  ||  23  i  1227- 


47- 
47. 


097160   ||   24  i  330445150    ||    25  |  36742802152    ||    26 


350152703494  ||  27  |  47190263648  ||  28  |  6119311582584 


ANS\VKH>.  319 


47.  ||J^jJ981473412'.Mi4  ||  30  j  T'.iSl  74437891 74_|_8i      36- 

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91  123 


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136.  ||  16  |  12.43  ||  17  |  59.827  ||   18  |  2223.1465   |   19  |  65- 

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15 

1  15  II   16  | 

25.5  1   17  |  9   ||    18  |  20   | 

19  |  '32.5 

139. 

20 

1  16  ||    21 

.60    1    22   |  3    ||   23  |  112 

|    24  |   12 

140. 

1  1 

12.50  ||  2  | 

62.50  ||  3  |  3.4375  ||  4  |  81.25 

1  5  |  22.5 

140. 

6  | 

415    ||     7 

|    45.33333£     ||     141.    ||    1 

412.875 

141. 

2  i 

3718.50  ||  i 

J  |  9984.50  [  4  |  27631.50  ||  f> 

|  TioT.f.G 

141. 

6  1 

2977.26  || 

1  |  21.40    ||    2  |  60.14£   1    3 

i   (i.  11712 

14-2. 

4  1 

8.40  ||  5  | 

164.8115    1   6  |  7654.90J  ||   1 

|  5.96124 

142J  2  |  23.597  ||  3  [  98.720875  ||  4  |  7.341 25  .jj  5  j  139.1)07/6 
143.  |[  1  |,75.385    ||   2  |  25.25    ||    3   |  66.666C.()|    ||    4  |   2.50 


143.  ||  1     91.87096+   |    2      96    |[    3  |   16    ||    144.  ||   1  |  025 


2  |  5.25,  7.00,  13.78125  ||  3  j  16.000816+;  64.65194  + 


144||  48050  ||   5  |  .06   ||   6  |  lOvod  ;   18.85174f 

144.  ||  7  |  .1875  ||  8  |  414.75  [|  9  |  3480  yd.;  4.50  per  yard. 

144.  i]  10  |  2.33333|  ||  11  |  11000,  w.'s;  5500,  ch.'s  ||  12  |  23.16 

145.  I  13  |  155  ||   14  |  547.92   ||   15  |  916  |   16  |  1,  100,TOO 


145.  ||  17  |  122.76616J    ||  I8_mj_l^j  15.68f°    |  20  |  92 

146.  "I  21  |  19.8  I  22  |  104.126  ||  23  |  27.685   ||   24  |  290.82 
146.  I  25  |  90277.70  ||  151.  ||  2  |  30183  ||  3  |  84226  |  4  |  391679 
151.  ||  5  |  84    ||    6   |  £1   12s.  3d.  Ifar.    ||    7   |  £25   14s.  Id. 


153.  1  3  |  316767   ||   4  |  359  mi.  7  fur.  28  rd.  ||  5  |~3796602 


153.  ||  6  |  8201  I  7     240700858  ||  8  |  109°  21|  mi.  7  fur.  1  rd. 


153.  1  3|  yd.  2  ft.  8  in.,  or  109°  22  mi.  3  fur.  1  rd.  4  yd.  1  ft.  2  in. 
I53T||~  9  |  4744  ||  10  |  5ft.  2 in.  ||  1  |  5ch.  601.~f "2  |~2682ft. 


153.  ||  10.8  in.  ||  3     172  ch.  581.  ||  154.  ||  1  |  575  |j  2  |  35yd. 


154.  ||  3qr.  3na.  ||  3  |  980  ||  4  |  623  ||  5     204  yd.  3  qr.  2  na. 


326  ANSWERS. 


154.  ||  6  |  28  E.F1.  1  qr.  ||  7  |  95  E.E.  4  qr.  ||   157.  ||  3  |  3157 
157.4     762300         5        260  >         6~~93  A.    2  R.    12  P. 


___ 

157.  ||  7  I  3~5mT~5~63  A.  1  R.  19  P.  ||  8  ]  12584.25  ||  9  |  15.25 
159.  ||  "3  "I  5927^TT^W(rfM^O~0"^ft-T5  I  5  C-  2  &&- 


1597||  6  |  2~T870  C.  40  ft.  ||  7  |  204  T.  11  cu.  ft.  1292  cu.  in. 


160.  ||  3     12002  ||  4  j  10  T.  1  pi.  ||  5  |  25  T.  1  gal.  ||  6     36.64 


161.  ||  3  |  23808  ||  4  |  844  ||_5J_272_|_6  |  Icb.  29 bu.  3pk.  6qt. 
163.  i|  3  |  2790366  |  4  |  903186  J  5  |  5  T.  8  cwt.  3  qr.  24  Ib. 


163.  ||  13  oz.  14  dr.  J  6  28  T.  4  cwt.  1  qr.  21  Ib.  ||  7  |  6  T. 


163.  [  2  cwt.  4  Ib.  13  oz.  14  dr.  |  8  |  2998128  ||  9  |  212  T. 
163711  14  cwt.  Iqr.  7  Ib.  ||  10  |  118.995-$10  ||  11  |  431.68-160 
1647]|  3  |  148340  ||  4  |  1  Ib.  1  oz.  lOpwt.  10  gr.  [  5  j  25  Ib. 
1 64.~f9  oz.  0 :p"wt~20"gr7'f  "6~i~6786l8~|j  7  |  36l'th  Toz7l4  pwt. 


164.  || 

8 

38901    ||    9  |   6496    |    10  |   657   ||    165.  ||  3  |  8011 

165.  || 

4 

T" 

9113  ||  5 

27  Ib  9  §  63  1  9  ||    6     94  Ib  11  §  1  3 

165.  || 

731)18  1  8 

i  12  Ib  9§  73  29  18  gr.  1  167.  1  3  |  379- 

168.  1 

467108  -||  4  ! 

4  yr.  1  da.  ||  5  |  24  yr.  1  da.  26  m.  58  sec. 

168.  || 

6 

9yr.  14  da, 

17  hr.  16m.  45sec.  ||  7  |  6600  ||  1    10765 

168.  || 

2 

2592000   | 

3     7°  44'  54"  ||  4  |  1  c.  5s.  28°   15' 

7j2°_23M)//  |  170.  ||  1  |  62208 
170.  1  2  I  $24  I  3  |  672  ca.  ft.— 5 J  cords.  |  4~   28. 1 25  "  |  5  | 


170.  ||   13165$,  13165J|f  ||  6    172.96875  ||  7  |  36288  ||  8  |  336 


170.  ||  9  |  223  ||  10  |  23856  ||  11  |  .88  loss.  ||  12  |  .6448  A- 
170.  I  48.4809  ||  13  |  14.34802+  |  14  58097  j  15  |  7  s.  155 
170.  ||  24'  40"  I  16  |  12  ||  17  |  1244T60~||"l8T3456  fW\ 
20  |  48976]  21  ("478602432  ]|  22  |  84  mi.  3 fur. 


170.  S  4  rd.  3>d.  2  ft.  ]|~23  |  5  A.  3  R.  35  P.  3  J  yd.  2  ft.  5  in. 


171.  ||  24  |  26880  ||  25     116280  j  26  |  27  ||  27  |  4  ||  28  |  40 


171.  ||  29  |  23  wk.  5  da.  16  hr.  ||  30  |  576  ||  31  |  110592  j_32  | 

mT 

172.  ||  270.0000  ||  12  |  1071.468  ||  13  |  992.4480  ||  14  i  559.92 


173.  ||  2  j  3f5-  ||  3  |  Tf 7  i|  4  |  T,j.0-o  ||  5  |  rf-s  ||  6     .0390625 


ANSWERS.  327 


173.  ||  7  |  .000560781|-   ||    8  |  .00409375    ||    9  |  .00007957  + 

1~73.  ||  10  |  1.74484375  [  11  |  .0496814+  ||  12  |  .000146484| 

173.  ||  13  |  .02734375    ||    14    |   .00035    ||    15    |   .0097222  + 

173.  I  16  j  1.3125     ||     17    |    71.15136+    ||     18   ]   .0039~68T 

1747]  1  |  1  pi.  1  lihd.  31  gal.  2  qt.    ||    2  |   3  qr.  2  na.  0.9  in. 


174.  ||  3     3wk.  Ida.  9  hr.  3(5  ra.  ||  4  |  13bu.  2pk.  ||  5  |  6  i'ur. 


174.  1  8  rd.  4  yd.  2  ft.  8  in.  |  6  |  3  cwt.  0  qr.  12  Ib.  8  oz. 
174.  ||  7  |  2  da.  13  hr.  42m.  51  f  sec.  ||  8  |  2bu.  2  pk.  ||  9  j  r.O 
174.  |galTTqt..  1  pt.  0|  ^[ToT"1  q^  21  lb-  10  oz-  ™%  (lr- 
174.  |  11  |  2gal.  3}  gi.  ||  12  |  2  R.  6  P.  4  ydT'o  ft.  127fV  in! 
174.  ~||T3j"l5cwtT3q^31b.  "i5oz."2"f'g:dr  ||'U  |  342da."4lir30Tn. 


175.  ||   1  |  12.00384  gr.  ||  2  |  2qr.  12  oz.  8  dr.  ||  3  |  2qt.Tpl 

.  3qr.  ||  6  |      8  P.     ||  7  |  1  hluf 


__ 

175T|i~4"T  gal,  rqtr'l"  TJTgal.'S  qt.   ||  !Tp36  da.  21  hr. 
imj  10  |  Is.  8d."IffarT  |~1  1  |  3qr.  lllb.  ||  12  |  lqr.71b.4oz. 
17(1  ||   13  |  19  hr.  40m.  48  sec.  ||  14  |  1  mi.  28rd.2yd.lft.lT7()4  iiu 
176.  j  T5~JToz.~¥dr.  1  16  |  103  da.  i%hr.  f»y"mTl2!4s 
niTlfTi  |  jBl.  Os.  lid.  oT6far.   ||    18  |'"^'iT7T~7idro."8far'. 

176.  ||  1  |  |  ||  2  |  Ty   ||  3  |  Ty   ||  4  I  A    I  5  I  I    I   6  I  j|-S§ 

177.  ||  7JJ5\9o   II  8  |  HI    II  -i_|_i^i_I  10  lit  11  |  T^/S 

177.  ||   1^  |  {-J-  li  13  I  -aiV2«.  ||  1  I  4.8899553+  ||  2  [  2.46944  + 

178.  I  3JT25  f  4~|  1.046875  g'~5     5.0"83331  ||  6  |  4.765625 


178.  |  7  |  .47291|   ||   8  |  .78875  ||   9  |  5.88125    ||    10  |  .0055 


178.  |   11     .42859226+       12      3920188         13      7.878125 


178.  ||   14  |  .778515625  ||   15  |  .1.5378472+   ||    179.  |  2  |  931 

179.  ||  Ib.  6  oz.  9  pwt.  5*f  gr.  ||  3  |  104  ib  3  I  3  3  2  3  4  gr. 
179T ||  4  |  254  T.  19  cw^TgrTTTbT^^  oz.  ||  5  |  50  T.  0  pT. 

179.  ||   1  hhd.  38  gal.  3  qt.    ||    6   |    138  ch.  30  bn.  3  pk.  5  qt 

180.  ||  7  I  172  yr.  2  mo.  1  wk.  4  da.  5^1  hr.  ||  8     29  s.  28°~32' 


180.  ||  49"  1  1  2^1  Ib-^oz.  15  pwt.  22  gr.  ||  2  |  432  L.  2  mi. 
180711  Tfar.  "39  rd.  4yd.  2gy  ft.  |  3  |  424  E.  FL  0  qr.  3  na. 
180.  ||  41  176cu.yd.  IScu.ft.  614ca.in.  |[  5  |  27  sq.mi.  2771T. 
f80.  ||  lR._OPL24iyd^||  6  |  244Tbr5iz."4^t._3gr.  |  7J_82T. 
180.  j  16  cwt.  16  Ib.  I  ozr'7'dr.  ||  8~|  41  T.  3  qr."l7  JbTsTdr! 


328  ANSWERS. 


181.  ||  9  |  336  A.  1R.  31  P.  210  sq.ft.  136sq.in.  ||  10  |  170T. 


181.  ||  11 

cu.  ft.  744  cu.in.  ||  11 

j  1  68  bu.  0  pk.  2  qt, 

||  12  |  45  A. 

181.  |i  3R.  31  P.  38  sq.ft.  ISOsq.in. 

||   13  |  158bu.  Opk.  4qt. 

181.  ||  14 

|  2  T.   5  cwt.  2  qr.   21  lb. 

i|    15    |    85yd.    ||     16    | 

181.  I  31b.  loz.  11  pwt.  17  gr.  | 

17  | 

322  mi.  6  fur. 

11  rd.  1  18  i 

182.  ||  100  A.  1R,  13 

P    II   1  i 

12  cwt.  Iqr.  7  lb.  13oz.  llf  dr. 

182.  1  2  | 

7  fur.  2  ft.  9  in.  ||  3  | 

1  mi. 

3  fur.  18rd. 

||  4  |  1  cwt. 

182.  ||  2  qr.  2  lb.  13  oz.    ||    5 

5  da.  20  hr.  52  m.  15}*  sec. 

182.  ||  6  | 

16s.  3d.  3.9far.  (  7  | 

6  cwt.  3  qr.  21  lb. 

5  oz.  8  dr. 

183.  ||  8 

56yd.  |j  9 

75  bu.  Opk.  11  qt.  [   10  |  90  mi.  4  fur. 

183.  ||  15rd.  lyd.  O.ft 

.  Hi  in. 

II   n 

2  yd.  2  qr.  1 

i-na.  J   12 

184.  ||  1  cwt.  1  qr.  7  lb.  7  oz. 

TVs-  dr.  ||  1  |  3  A. 

0  R.  38  P. 

184.  1  2  | 

IT.  14  cwt. 

Oqr.  19  lb.  ||  3 

|  175  lb.  loz. 

1  pwt.  3  gr. 

184.  ||  4 

81b.  lOoz.  14  pwt.  4gr.  || 

5  |  5  T.  7  cwt.  1  qr.  23  lb. 

184.  1   11 

oz.  I    6  |  7  cwt.  2  qr. 

20  lb 

1  1  oz.  5  dr. 

7  |  124  T. 

184.  ||  Ohhd.  59  gal.  || 

8  |  14  yr. 

46  wk.  4.  da.  20  hr.  58m.  54  sec. 

184.  1  9 

14  mi.  7  fur,  37  rd.  2  yd.  2  ft.  9  in.  ||   10 

46  A.  3R. 

185.  i|  35 

P.  13yd.  8| 

9  ft     II     1  1 
6  "•    II     l  l 

|  £5  17s.  6fd.  ||   12  |  Is.  24° 

185.  ||   19 

32Ty  ||    13   |   27  mo.  3  wk.  0  da.  2-0 

hr.  20  min. 

185.  j|   14 

,    84  yr.    11  mo.    1 

wk.    5  da.    1     15 

|    £2    17s. 

185.  ||  16 

|  1  lb.  11  oz.  19  pwt.  4  gr.  ||   17  |  6  ft 

10  3  53  13 

185.  ||  18 

|  7  T.I  8  cwt 

lqr.41b. 

Ooz.2dr.  ||  19|2mi 

.4fur.21rd. 

185.  ||  20 

68  lb.  lOoz.  3  pwt. 

15  gr. 

||  21  |  IT.  17  cwt.  3qr. 

185.  1  7  lb.    14  oz.   2 

dr.     ||     22    |    84  ft   9  !   4  3 

1  3    14  gr. 

185.  ||  23 

3  yd.   2  qr.    1  na.   j 

f  in.  I 

24  |  4  C.  3  C.  ft.  2  cu.  ft. 

188.  ||  2  | 

9  yr.  4  mo.  2  da,  |  3 

21  yr.  9  mo.  5  da. 

II  4      17  yr. 

186.  ||  Imo.  3  da.  |  6 

|  1  2  yr.  3  mo. 

26  da,  22  hr. 

II   7  |30yr. 

186.  1   Imo.  29  da.  12 

hr.  ||  8  | 

7yr. 

9  mo.  1  da.   [ 

9  |  369  yr. 

187.  ||  9  mo.  14  da.    | 

1    |    6  pwt.  15  gr.    ||    2  | 

£1   9s.  3d. 

3  gr.  |  4  |  11  hr.  59  m.  59-£-  sec! 
187.  j'  5  |  3  yd.  2  ft." "|'|"  6~j  J~6  gal.  2  qtT  0  pt.  2|f  gi.  !j  7~i 
187.  ||  U  pwt.  3  gr.  I  8  |  4  cwt.  1  qr.  12  lb.  15  oz.  S^'df. 
187.  ||  9  J_8cwt.  3qr.  51b.  13oz.  Of  5- dr.  |  10  j  311>.  5oz.  16p\vt". 
187T  Illgr.  ||  11  |  Ird.  lyd.  2ft.~5J~in.  ||  12 •  |  7§  5g  23  lOgr! 


ANSWERS.  329 


187.  ||  13  |  2  hhd.  25  gal.  3  qt.  0  pt.  0.292  gi.   ||    14  |  14  da. 

187.  ||  19   hr.    15  min.    58.464   sec.     ||     15  ~| 

1897||  3  |   56  mi.    5  furTTrd.    ||    T~\~W~B.  45^ 

189.  |i  5  I  32  yr.  3  mo.  18  da,  18  hr.  ||  6  |  5^  T-_3jc\vt  2 
i89Tl"T6"lb.  4oz.  8"drTi"8  |  25  bu.  3pk.  l~qt.  jj  »  \  s  ('.  6 

189.  I  10  |  17  yr.  5  mo.  a  (}ar||TTflTlO  oz.  I2j>wt.  j, 

"189.  ||  1  T.  19  cwt    2  qr.  12"lbT|ri3~|  13  fc  75  23134  gr. 


190.  ||  14  |  122  rni.  4  fur.  23J  rd.  ||   15      111  A.  2  II.  25  P. 


190.  ||   16     267  yd.  0  qr.  3  na.  ||  17  |  47  L.  1  mi.  7  fur.  8  rd. 
190.  ||   18    "95  hhd.  6  gal  |   19  |~32~Ib.  9  oz.  15  pwt.  ||  20~| 


190.  ||  746  mi.  5  fur.  ||  21  |  15°  ||  22  |  50  T.  14  cwt.  3  qr.  15  Ib. 
19071  23  |  £5  4s.  3d.  ||  24  [  24  hhd.  22  gal.  1  qt.  1  pt.  ||  25  | 


191. 

927 

yd.| 

2  |  12  A.  2 

R.  25  P.  ||  3  |  5  L.  2  mi.  6  fur.  36  rd. 

192. 

4| 

2  bu. 

3  pk.  4  qt. 

||    5   |   2  cwt.  1  qr.  18  Ib.  3 

192. 

11 

5  yd.  \ 

2  qr.  OJ  na. 

||  7  |  251b.3oz.8dr.  ||  8  |  •_ 

192. 

16' 

I  9 

49  gal.  2 

qt.  1  pt.  ||   10  |  5  bu.  1  pk.  6J-  qt. 

192. 

11 

3  fo 

4  I  63  1 

9  16  gr.  ||   12  |  12  A.  2  R.  25  P. 

192. 

13 

2T. 

7  cwt.  ||  14 

61  gal.  IqtTlpt.  ||  15  |  £1  2s.  4£d. 

192J|  16  |  £21  9s.  8d.  |  17  |  1  pk.  2  qt.  0  pt.  2.95258T  gi. 
192711  "18  [  17  cwt.  3  qr.  18  Ib.  12  oz.  2f  |  dr.  ||  19  [  2  bbd. 
192.  ||  54  gal.  3  qt.  1.65  pt.  ||  20  |  24  mi.  7  fur.  4  rd. 
192711  21  |  14  Ib.  0  oz.  8  pwt.  11  gr.  ||  22  |  3  gal.  1.5  qt. 
1937  1|  1T^6~A7"3  R.  22  j  P.  J  2  |  4  bu.  3  pk7"2"qt". 
.  7  mo.  3  wk.  2  da.  8  hr.  |  4  |  2  cwt.  1  qr.  24~lb. 
1.73026+  oz.  ||  6  |  15'  ||  7  |  2  bu.  7~qt. 
gal.  2qt.  0.62711+  pt.  ||  9  '  ~ 


193T||T2"  cwt.  2  qr.  11  Ib.     |    11    |    124  mi.  3  fur.  20  rd. 
193.  ||  12  |  4^   ||  13  |  326  j  14  |  24  reams  5  quires  12  sheets 


194.  1   15      10H    ||     16     |    £63    14s.   8d.    |     17 


194.  ||  18  |  Ihhd.  19  gal.  0  qt.  1  pt.  ||    19  |  222jgf  =  222-g| 

194.  [I  20  |  61  mi.  5 fur.  29 rd.  ||  21  |  20 T.  3 cwt.  Iqr.  13.1 25 Ib. 

194.  II  22  |  $3795,  24  T.  0  cwt.  2  qr.  20 Ib.  ||  23  |  S^pT  |  88^ 

196.  [  2  |  llhr.  55  m.  24  sec.  A.M.  |  3  |  llhr.  18m.  28secTIjtf. 

19'6.  I  4  j  lOhr.  59m.  5 6 sec.  A.M.  ||  5  |  2  hr.  20 HI.  3|  sec.  p.  M. 


330 


ANSWERS. 


197.  ||  6  | 

6  hr.  0  m.  8  sec.  A 

M.  I  7 

1   12 

hr.  53  m.  56 

sec.  P.  M 

197.  ||  2  | 

78°  52'  W.  ||   3 

TO 

0  37 

'  W.-  ||   4 

|  74°  1 

/  2" 

W 

197.  ||  5  | 

33°  15'30"W.  | 

199.  1  1 

|  15ft.  5' 

12 

|  1ft.  8' 

10' 

199.  ||  34 

2ft,  6'  3"  11'"  | 

4 

5ft.  8' 

2"  1' 

"  II 

5 

15ft.  4 

199.  ||   10"  4'"  ||    6  |  1ft.  11 

'  10"  11'" 

II    ?| 

11 

ft.  6'  5" 

5" 

199.  ||  8  | 

7ft.  10;  i"  9'"  I 

9  I  f  II 

10 

i    89    y  i  i  i 

1    T08-    l_*       I    ' 

36979  + 

199.  ||  12 

|  .0520831  ||  201 

.  ]   1  |  77  sq. 

ft.  || 

2| 

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1   13 

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|  May  2d   ||   3  |  51  days—  Jan.  26th 

260.  ||  4  | 

Jan.    30th, 

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335 


301.  || 

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311.  I 

4 

4717    ||    5   |   137    ||    6   |   f    1    7    |   17J    ||    8   |   yfc 

311.  1 

9 

4/T  ||   10  |  9.04  1   11  |  41   ||    12  |   978    ||    13  |  51 

311.  || 

14     80.71     |i     312.    ||     15     |     5467     ||     16     |     26.25 

312.  || 

17     213.3125;   211.6875    |    18   |  ff  ;   1J    |    19   |   50 

312.  || 

20     879000  ||  21  |  792  ||  22  |  1  mi.  6  fur.  33rd.  15^  ft. 

312.  || 

23     3567  1  24  |  108Y75  ||  25'  |  8(J  j  26  |  120  ||  27  |  3800 

312.  || 

28     2400    ||    29  |  28;    60;    32    ||    30  |  2364;    1182; 

313.  1 

788;    394    ||    31  |  7£    ||    32  |  3    ||    33  |  ^  ;    986.66§ 

313.  || 

34     11   ||    35  |  160;    120;    140    ||    36  |  2454;    3681; 

313.  1 

4294.50   ||    37  |  408    ||   38  |  62    ||    39  |  4    ||   40  |  288 

313.  || 

41  |  23|  ||  42  |  2250  |   314.  ||  43  |  22500  |    44  j  864 

:U4.  ||  45     300  1  46  |    57    ||  47  |  52  yr.  11  mo.  20  da.  10£hr. 

336 


ANSWERS. 


3M. 

314: 

314. 
314. 
315. 
315. 
315. 

3Ts: 


48  ;  111  |j  49  | 
52  I  3.6538+ 


50  |  32T8r  m.  ||  51  |  34782.60869 
~ 


53 


337 


54         7816.09195 


55 


129.87?V  ;   97.40ff  ;   77.92jf 


||  56(920.20;    2760.60;   5521.20    ||    315.   ||    57 


58  3  hr.  20  min.    59   27.78084 


|    240 
5  yr. 


61  j  196.88  ;  295.24 


96  :     12  ;     12 


I  63  |  i6}f   I    64   |   356.25    ||    65   |   8640   ||    66   |   4 


yr. 


11  mo.  27+ da.   ||   67  |  1020.66    ||    316.  ||    68  |  8925- 
T544T1   69  |  1.20   I    70  |  4    ||    71  ]~T|f~72T4~yd~. 


316. 
316~ 


73  |  423.36  ||  74  |  50  ||  75  |  780;  3120;  1560  ||76| 


18.5+  I  77  |  13,  or  A  was  27  ||  78 


79  |  21; 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

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on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


ICLF  ( 

N)  ' 

•  ^s  •"      " 

•  n  / 

JAN  2  01966  6 

REC'U  LD 

O  nil 

m   fe'66'8PW 

SENT  ON  ILL 

MAY  2  7  1994 

U.  C.  BERKELEY 

L&£3S$i£             u-S^gSU. 

17360 


cStancW  School 


5I&.  :  ^  JOHN  ST.  N.Y 


1.    SPELLIN  -,  HTML-ING,  AND   ELOCUTION. 


BY   R.  O.  :  AK 

The  National  Sch    >1  P-imer. 
•ional  Fir.M,  1: 

:::>a!  Sero-id  leader, 
tional  Thir-.l  keader. 
The  Nat;  ; leader. 

ifth  Header. 

•..ionai  Pronouncing  Speller. 
'Ihe  National  Klenu-ntary  SpJIer. 


3.  MADISON    WATSON. 

Parker's  i;heti>rical  Header. 
Smith's  Juvenile  Definer. 
Smith's  Grammar-School  Spoiler. 
Smith's  Detnier's  Manual. 
Wright's  Analytical  Orthograj)hy. 
Day's  Art  of  Klo.-ntion. 
High  School  Literati!!-- 
Brooks'  School  Manual  of  Devotion. 


2.    ENGLISH  GRAMMAR,  RHETORIC,   &c. 


Cinrk's  First  Lvs.sons  in  Grammar. 
Clark's  New  English  Granumir. 
Clark's  Analysis  Of  the  English  !>.: 
AVclcb'd  Anu'lysis  <>f  is 
Muh.-in'.s  S:-ii»iiw  of  Lo-ric,  ' 

j  hy. 
J)ny'8  Art  of  Uhftoric. 

3.    MONTEITH  ANt 

Mo:>! Dili's  First.  Lessons  h 


"WillanTs  Morals  f«r  the  Yo... 

"'.lement.s  of  Crit  ici> 
Boyd's  Milton's  Piiradise  L.-st. 

of  Time. 
.  itc. 
Boyd's  '• 

i  liongbiv 

a  SEEIES  OF  GEOGRAi»HI 

O^ER  MAT 
gebra, 

iiK'try. 

-ml  I  nti-ijral  C 

•uK-try. 

Davies'  SI 

Davit-  .-..-try  of 


UVIES'  SERIL 

iinar  of  Arum 

lira. 

I     IXavies'  Elementary  Geometry. 
I     Davies'  Practice!  Mathen.  . 
vies'  University  Algebra. 

TORY  AND  MYTHOLOGY,*  &c. 
I     MontoHli's  Youth's  Hist,  of  Ui  i 
1's  School  Hist,  of  tii,- 
r»  Large  Hisi.  of  th. 
:  il's  Universal  History  JM 

6.  scnafT.:Fic 

Parker's  Juvenile  ] 

•'•'s  Juvenile  Philo 
•:-'s  Natural  P^ 

•^  Book  of  Ci 

Pack's  Elements   .1 
Darby's  Southern  Bo; any. 

7. 

Brooks'  First  Lat'n  r. 
Brooks'  First  G:vok  : 
Brook»'  Co'lectunea  Evangr-'.1 

TUB  S, 


v  •  with  Kn 

-tory  of  iho  . 

:,  Myti, 

DEPARTMENT. 

|  Melntyreo- 

nomy. 

•try. 
•.-nay's  Elements  of  Cale'ulns. 


:K3'   CLASSICS. 
Broo) 

lir.-o'K-;'  Cresar,  with  Ili-istn»tH)n9. 
HKOOKS'  SCHOOL  TKACHKK 


